Cadeias de opções binárias

lxml FAQ - Perguntas Frequentes.
Perguntas frequentes sobre lxml. Veja também as notas sobre compatibilidade com o ElementTree.
Questões gerais.
Existe um tutorial?
Leia o tutorial lxml. etree. Embora isso ainda esteja em andamento (assim como qualquer boa documentação), ele fornece uma visão geral dos conceitos mais importantes em lxml. etree. Se você quiser ajudar, melhorar o tutorial é um ótimo lugar para começar.
Há também um tutorial para o ElementTree, que funciona para o lxml. etree. A documentação da API estendida do etree também contém muitos exemplos para o lxml. etree. A biblioteca de elementos de Fredrik Lundh contém muitas receitas interessantes que mostram como resolver tarefas comuns em ElementTree e lxml. etree. Para aprender usando lxml. objectify, leia a documentação do objectify.
John Shipman escreveu outro tutorial chamado processamento XML Python com lxml que contém muitos exemplos. Liza Daly escreveu um belo artigo sobre aspectos de alto desempenho ao analisar arquivos grandes com lxml.
Onde posso encontrar mais documentação sobre o lxml?
Há muita documentação na Web e também na documentação da biblioteca padrão do Python, já que o lxml implementa a bem conhecida API ElementTree e tenta seguir sua documentação da melhor forma possível. Vale a pena dar uma olhada nas receitas da biblioteca de elementos de Fredrik Lundh. Existem alguns problemas em que o lxml não consegue manter a compatibilidade. Eles estão descritos na documentação de compatibilidade.
As extensões específicas do lxml para a API são descritas por arquivos individuais no diretório doc da distribuição de origem e na página da web.
A documentação da API gerada é uma referência abrangente da API para o pacote lxml.
Quais padrões a lxml implementa?
A conformidade com os Padrões XML depende do suporte em libxml2 e libxslt. Aqui está uma citação do xmlsoft /:
Na maioria dos casos, a libxml2 tenta implementar as especificações de maneira relativamente rígida. A partir do release 2.4.16, a libxml2 passou todos os 1800+ testes do OASIS XML Tests Suite.
O lxml suporta atualmente o libxml2 2.6.20 ou posterior, que tem suporte ainda melhor para vários padrões XML. Os mais importantes são:
O suporte para XML Schema não está 100% completo no libxml2, mas é definitivamente muito próximo da conformidade. O Schematron é suportado de duas maneiras, sendo o melhor a implementação de referência original do ISO Schematron via XSLT. A libxml2 também suporta o carregamento de documentos através de HTTP e FTP.
Para suporte ao RelaxNG Compact Syntax, existe uma ferramenta chamada rnc2rng, escrita por David Mertz, que você pode usar no Python. Caso contrário, trang é a ferramenta de linha de comando 'oficial' (escrita em Java) para fazer a conversão.
Quem usa lxml?
Como uma biblioteca XML, o lxml é frequentemente usado em aplicativos de servidor internos, como servidores da Web ou aplicativos que facilitam algum tipo de gerenciamento de conteúdo. Muitas pessoas que implementam o Zope, o Plone ou o Django o usam junto com o lxml em segundo plano, sem falar publicamente sobre ele. Portanto, é difícil ter uma ideia de quem a utiliza e a seguinte lista de 'usuários e projetos que conhecemos' está muito longe de ser uma lista completa de usuários do lxml.
Observe também que a compatibilidade com a biblioteca ElementTree não requer que os projetos definam uma dependência rígida em lxml - contanto que eles não aproveitem o conjunto de recursos aprimorado do lxml.
cssutils, um analisador de CSS e kit de ferramentas, pode ser usado com lxml. cssselect Deliverance, uma ferramenta de tema de conteúdo Enfold Proxy 4, um acelerador de servidor da Web com processamento XSLT em tempo real Inteproxy, um proxy HTTP seguro lwebstring, um mecanismo de modelo XML openpyxl , uma biblioteca para ler / gravar arquivos do MS Excel 2007 OpenXMLlib, uma biblioteca para manipular metadados de documentos OpenXML PsychoPy, software de psicologia no Python Pycoon, uma estrutura de desenvolvimento web WSGI baseada em pipelines XML pycsw, uma implementação de servidor OGC CSW escrita em Python PyQuery, um framework de consulta para XML / HTML, semelhante ao jQuery para JavaScript python-docx, um pacote para lidar com o formato Rambler do Word OpenXML da Microsoft, um mecanismo de busca que agrega diferentes fontes de dados rdfadict, um analisador RDFa com uma interface simples semelhante a um dicionário. xupdate-processor, uma implementação do XUpdate para o lxml. etree Diazo, um motor de temas do site XSLT sob o capô.
O Zope3 e algumas de suas extensões têm um bom suporte para lxml:
gocept. lxml, ligações de interface Zope3 para lxml z3c. rml, uma implementação do formato RML do ReportLab zif. sedna, uma interface baseada em XQuery para o banco de dados XML Sedna OpenSource.
E não perca as citações de nossos usuários geralmente felizes e de outros sites vinculados a lxml. Como diz Liza Daly: "Muitos produtos de software vêm com a restrição de escolha, o que significa que você deve escolher apenas dois: velocidade, flexibilidade ou legibilidade. Quando usado com cuidado, o lxml pode fornecer todos os três."
Qual é a diferença entre lxml. etree e lxml. objectify?
Os dois módulos fornecem diferentes maneiras de lidar com XML. No entanto, objetivamos construir sobre lxml. etree e, portanto, herdar a maioria de seus recursos e uma grande parte de sua API.
O lxml. etree é uma API genérica para manipulação de XML e HTML. Ele tem como objetivo a compatibilidade com o ElementTree e suporta todo o infoset XML. É bem adequado para conteúdo misto e XML centralizado em dados. Sua generalidade faz com que seja a melhor escolha para a maioria das aplicações.
lxml. objectify é uma API especializada para manipulação de dados XML em uma sintaxe de objeto Python. Ele fornece uma maneira muito natural de lidar com campos de dados armazenados em um formato XML estruturalmente bem definido. Os dados são convertidos automaticamente em tipos de dados do Python e podem ser manipulados com operadores normais do Python. Veja os exemplos na documentação do objeto para ver como é usá-lo.
O Objectify não é adequado para conteúdo misto ou documentos HTML. Como ele é construído sobre o lxml. etree, no entanto, ele herda o suporte normal para XPath, XSLT ou validação.
Como posso fazer meu aplicativo ser executado mais rapidamente?
O lxml. etree é uma biblioteca muito rápida para processar XML. Existem, no entanto, algumas ressalvas envolvidas no mapeamento da poderosa biblioteca libxml2 para a simples e conveniente API ElementTree. Nem todas as operações são tão rápidas quanto a simplicidade da API pode sugerir, enquanto alguns casos de uso podem se beneficiar da maneira correta de realizá-los. A página de referência tem uma comparação com outras implementações do ElementTree e várias dicas para ajustes de desempenho. Como acontece com qualquer aplicativo Python, a regra básica é: quanto mais o processamento for executado em C, mais rápido será o aplicativo. Veja também a seção sobre segmentação.
E quanto ao texto à direita em elementos serializados?
O modelo de árvore ElementTree define um Elemento como um contêiner com um nome de tag, texto contido, Elementos filho e um texto final. Isto significa que sempre que você serializar um elemento, você obterá todas as partes desse elemento:
Aqui está um exemplo que mostra porque não serializar a cauda seria ainda mais surpreendente do ponto de vista de um objeto:
Imagine uma lista do Python onde você acrescenta um item e ele não aparece quando você olha para a lista.
A propriedade. tail é uma grande simplificação para o modelo de árvore, pois evita que os nós de texto apareçam na lista de filhos e torna o acesso a eles rápido e simples. Portanto, isso é um benefício na maioria dos aplicativos e simplifica muitos algoritmos de árvore XML.
No entanto, em XML semelhante a documento (e especialmente HTML), o resultado acima pode ser inesperado para novos usuários e às vezes pode exigir um pouco mais de sobrecarga. Uma boa maneira de lidar com isso é usar funções auxiliares que copiem o elemento sem sua cauda. O pacote lxml. html também lida com isso em alguns lugares, já que a maioria dos algoritmos HTML se beneficia de um comportamento sem cauda.
Como posso descobrir se um elemento é um comentário ou PI?
Como posso mapear uma árvore XML em um ditto de ditos?
Estou feliz que você tenha perguntado.
Observe que esse belo conversor rápido e sujo espera que os filhos tenham nomes de tags exclusivos e sobrescreverão silenciosamente quaisquer dados contidos nos irmãos anteriores com o mesmo nome. Para qualquer aplicação real de conversão de xml para dict, é melhor escrever sua própria versão mais longa.
Por que o lxml às vezes retorna valores 'str' para texto no Python 2?
No Python 2, a API do lxml retorna seqüências de caracteres de bytes para valores de texto ASCII simples, seja para nomes de tags ou texto no conteúdo do elemento. Esse é o mesmo comportamento conhecido do ElementTree. O raciocínio é que as strings de bytes codificadas em ASCII são compatíveis com strings Unicode no Python 2, mas consomem menos memória (geralmente por um fator de 2 ou 4) e são mais rápidas de criar porque não requerem decodificação. Valores simples de string ASCII são muito comuns em XML, então essa otimização geralmente vale a pena.
No Python 3, o lxml sempre retorna strings Unicode para texto e nomes, como o ElementTree. Desde o Python 3.3, cadeias Unicode contendo apenas caracteres que podem ser codificados em ASCII ou Latin-1 são geralmente tão eficientes quanto as strings de bytes. Em versões mais antigas do Python 3, os inconvenientes mencionados acima se aplicam.
Instalação.
Qual versão de libxml2 e libxslt devo usar ou requerer?
Isso realmente depende do seu aplicativo, mas a regra básica é: versões mais recentes contêm menos bugs e fornecem mais recursos.
Não use libxml2 2.6.27 se você quiser usar XPath (incluindo XSLT). Você receberá falhas quando ocorrerem erros XPath durante a avaliação (por exemplo, para funções desconhecidas). Isso acontece dentro da chamada de avaliação para libxml2, então não há nada que o lxml possa fazer sobre isso. Tente usar versões de ambas as bibliotecas que foram lançadas juntas. Pelo menos a versão libxml2 não deve ser mais antiga que a versão libxslt. Se você usar o XML Schema ou o Schematron que ainda estão em desenvolvimento, a versão mais recente do libxml2 geralmente é uma boa aposta. O mesmo se aplica ao XPath, onde um número substancial de bugs e vazamentos de memória foram corrigidos ao longo do tempo. Se você encontrar falhas ou vazamentos de memória nos aplicativos XPath, tente uma versão mais recente do libxml2. Para analisar e corrigir HTML quebrado, o lxml requer pelo menos o libxml2 2.6.21. Para o manuseio normal da árvore, no entanto, qualquer versão da libxml2 começando com 2.6.20 deve ser feita.
Leia as notas de lançamento do libxml2 e as notas de lançamento do libxslt para ver quando (ou se) um bug específico foi corrigido.
Onde estão as construções binárias?
Graças à ajuda de Joar Wandborg, tentamos disponibilizar compilações binárias "manylinux" para Linux logo após cada release de origem, pois elas são usadas com muita frequência por integração contínua e / ou servidores de compilação.
Graças à ajuda de Maximilian Hils e do serviço de criação do Appveyor, também tentamos atender às frequentes solicitações de compilações binárias disponíveis para o Microsoft Windows em tempo hábil, já que os usuários dessa plataforma geralmente não criam lxml por conta própria. Dois dos principais problemas de design deste sistema operacional tornam isso não trivial para seus usuários: a falta de um compilador padrão pré-instalado e o gerenciamento de pacotes ausente.
Além disso, Christoph Gohlke fornece generosamente compilações binárias lxml não oficiais para o Windows que são geralmente muito atualizadas. Considere usá-los se você preferir uma compilação binária sobre uma versão oficial assinada.
Por que recebo erros sobre a falta de símbolos UCS4 ao instalar o lxml?
Você está usando uma instalação do Python que foi configurada para uma representação Unicode interna diferente do pacote lxml que está tentando instalar. As versões do CPython anteriores a 3.3 permitiam alternar entre dois tipos no tempo de compilação: a codificação de 32 bits UCS4 e a codificação de 16 bits UCS2. Infelizmente, ambos não são compatíveis, então os ovos e outras distribuições binárias só podem suportar o que foram compilados.
Isso significa que você precisa compilar o lxml de fontes para o seu sistema. Note que você não precisa do Cython para isso, a distribuição fonte lxml é diretamente compilável em ambos os tipos de plataforma. Veja as instruções de construção sobre como fazer isso.
Meu compilador C falha na instalação.
O lxml consiste em uma quantidade relativamente grande de código C gerado (Cython) em um único módulo de origem. A compilação deste módulo requer muita memória livre, normalmente mais de meio GB, o que pode causar problemas, especialmente em sistemas compartilhados / em nuvem.
Se o seu compilador C trava ao construir lxml a partir de fontes, considere o uso de uma das rodas binárias que fornecemos. Os binários "manylinux" geralmente devem funcionar bem na maioria dos sistemas de compilação e instalar substancialmente mais rápido que uma compilação de origem.
Contribuindo
Por que o lxml não está escrito em Python?
O lxml não é escrito em Python simples, porque ele faz interface com duas bibliotecas C: libxml2 e libxslt. Acessá-los no nível C é necessário por motivos de desempenho.
No entanto, para evitar escrever código C simples e se importar muito com os detalhes de tipos internos e contagem de referência, o lxml é escrito em Cython, um superconjunto da linguagem Python que se traduz em código-C. As chances são de que, se você conhece Python, pode escrever o código que o Cython aceita. Novamente, o estilo C-ish usado no código lxml é apenas para otimizações de desempenho. Se você quiser contribuir, não se preocupe com os detalhes, uma implementação Python de sua contribuição é melhor que nenhuma. E tenha em mente que a API flexível do lxml geralmente favorece a implementação de recursos em Python puro, sem se preocupar com o código-C. Por exemplo, o pacote lxml. html é escrito inteiramente em Python.
Por favor, entre em contato com a lista de discussão se precisar de ajuda.
Como posso contribuir?
Se você encontrar algo que gostaria que o lxml fizesse (ou melhor), por favor, nos fale sobre isso na lista de discussão. Pedidos de pull no github são sempre bem-vindos, especialmente quando acompanhados por testes de unidade e documentação (doctests seria ótimo). Consulte os subdiretórios de testes na árvore de origem lxml (abaixo do diretório src) e os arquivos de texto ReST no diretório doc.
Também temos uma lista de recursos ausentes que gostaríamos de implementar, mas não devido a falta de tempo. Se você encontrar o tempo, os patches são muito bem vindos.
Além de melhorar o código, há muitos lugares onde você pode ajudar o projeto e sua base de usuários. Você pode.
espalhe a palavra e escreva sobre lxml. Muitos usuários (especialmente novos usuários do Python) ainda não ouviram falar sobre o lxml, embora nossa base de usuários esteja crescendo constantemente. Se você escrever seu próprio blog e quiser dizer algo sobre lxml, vá em frente e faça isso. Se acharmos que sua contribuição ou crítica é valiosa para outros usuários, podemos até colocar um link ou uma cotação na página do projeto. forneça exemplos de código para o uso geral de lxml ou problemas específicos resolvidos com lxml. Código legível é uma ótima maneira de mostrar como uma biblioteca pode ser usada e quais são as grandes coisas que você pode fazer com ela. Novamente, se ouvirmos sobre isso, podemos definir um link na página do projeto. trabalhar na documentação. A página da web é gerada a partir de um conjunto de arquivos ReSTtext. Significa tanto como uma página de projeto representativa para o lxml e como um site para documentar a API e o uso do lxml. Se você tiver dúvidas ou uma idéia de como torná-lo mais legível e acessível enquanto estiver lendo, envie um comentário para a lista de discussão. melhorar o site. Colocamos algum trabalho em tornar o site útil, compreensível e também fácil de encontrar, mas sempre há coisas que podem ser feitas melhor. Você pode perceber que não estamos no topo do ranking quando pesquisamos na web por "Python e XML", então talvez você tenha uma ideia de como melhorar isso. ajuda com o tutorial. Um tutorial é o ponto de partida mais importante para novos usuários, por isso é importante para nós fornecer um guia fácil de entender em lxml. Como toda a documentação, o tutorial está em andamento, por isso agradecemos cada ajuda. melhorar os docstrings. O lxml usa docstrings para suportar a função de ajuda online integrada do Python (). No entanto, às vezes, eles não são suficientes para entender os detalhes da função em questão. Se você encontrar um lugar assim, você pode tentar escrever uma descrição melhor e enviá-lo para a lista de discussão.
Meu aplicativo falha!
Um dos objetivos do lxml é "sem segfaults", por isso, se não houver um aviso claro na documentação de que você estava fazendo algo potencialmente prejudicial, você encontrou um bug e gostaríamos de saber sobre ele. Por favor comunique este bug à lista de discussão. Veja a seção sobre relatórios de erros para aprender como fazer isso.
Se o seu aplicativo (ou, por exemplo, seu contêiner da web) usar encadeamentos, consulte a seção de perguntas frequentes sobre encadeamentos para verificar se você toca em uma das potenciais armadilhas.
Em qualquer caso, tente reproduzir o problema com as versões mais recentes de libxml2 e libxslt. De tempos em tempos, bugs e condições de corrida são encontrados nessas bibliotecas, portanto, uma versão mais recente já pode conter uma correção para seu problema.
Lembre-se: mesmo se você vir lxml aparecer em um rastreamento de pilha de falhas, não é necessariamente lxml que causou a falha.
Meu aplicativo falha no MacOS-X!
Este foi um problema comum até o lxml 2.1.x. Desde o lxml 2.2, a única forma oficialmente suportada de usá-lo nesta plataforma é através de uma compilação estática contra versões recentemente baixadas de libxml2 e libxslt. Veja as instruções de compilação para o MacOS-X.
Eu acho que encontrei um bug no lxml. O que devo fazer?
Primeiro, você deve examinar o log de alterações atual do desenvolvedor para ver se esse é um problema conhecido que já foi corrigido na ramificação principal desde o release que você está usando.
Além disso, a seção 'crash' acima tem alguns bons conselhos sobre o que tentar ver se o problema está realmente em lxml - e não em sua configuração. Acredite ou não, isso acontece com mais frequência do que você imagina, especialmente quando bibliotecas antigas ou até mesmo várias versões de bibliotecas são instaladas.
Você deve sempre tentar reproduzir o problema com as versões mais recentes de libxml2 e libxslt - e certifique-se de que elas sejam usadas. O lxml. etree pode dizer com o que é executado:
Se você puder descobrir que o problema não está em lxml, mas na libxml2 ou libxslt subjacente, você pode perguntar diretamente nas respectivas listas de discussão, o que pode reduzir consideravelmente o tempo para encontrar uma correção ou uma solução alternativa. Veja a próxima pergunta para algumas dicas sobre como fazer isso.
Caso contrário, nós realmente gostaríamos de ouvir sobre isso. Por favor, informe ao bug tracker ou à lista de discussão para que possamos corrigi-lo. Nesse caso, é muito útil se você puder criar um snippet de código curto que demonstre seu problema. Se outros puderem reproduzir e ver o problema, é muito mais fácil para eles consertá-lo - e talvez até mais fácil para você descrevê-lo e convencer as pessoas de que realmente é um problema para consertar.
É importante que você sempre relate a versão do lxml, libxml2 e libxslt obtida do snippet de código acima. Se não soubermos as versões da biblioteca que você está usando, perguntaremos de volta, portanto, será mais demorada para você obter uma resposta útil.
Já que como usuário de lxml você provavelmente é um programador, você pode achar este artigo sobre relatórios de erros uma leitura interessante.
Como eu sei que um bug está realmente em lxml e não em libxml2?
Uma grande parte da funcionalidade do lxml é implementada por libxml2 e libxslt, portanto, os problemas encontrados podem estar em um ou no outro. Saber o local certo para perguntar reduzirá o tempo necessário para solucionar o problema ou para encontrar uma solução alternativa.
Tanto libxml2 quanto libxslt vêm com seus próprios frontends de linha de comando, ou seja, xmllint e xsltproc. Se você encontrar problemas com o processamento de XSLT para folhas de estilo específicas ou com validação para esquemas específicos, tente executar o XSLT com xsltproc ou a validação com xmllint respectivamente para descobrir se ele também falha lá. Em caso afirmativo, por favor informe diretamente as listas de discussão do respectivo projeto, a saber:
Por outro lado, tudo o que parece estar relacionado ao código Python, incluindo resolvedores customizados, funções XPath customizadas, etc., provavelmente está fora do escopo da libxml2 / libxslt. Se você encontrar problemas aqui ou não tiver certeza de onde o problema pode vir, por favor, pergunte na lista de discussão lxml primeiro.
Em qualquer caso, uma boa explicação do problema, incluindo algum código de teste simples e alguns dados de entrada, nos ajudará (ou os desenvolvedores da libxml2) a ver e entender o problema, o que aumenta bastante sua chance de obter ajuda. Veja a pergunta acima para algumas dicas sobre o que é útil aqui.
Posso usar encadeamentos para acessar simultaneamente a API lxml?
Resposta curta: sim, se você usar o lxml 2.2 e posterior.
Desde a versão 1.1, o lxml libera internamente o GIL (bloqueio global do interpretador do Python) ao analisar o disco e a memória, desde que você use o analisador padrão (que é replicado para cada thread) ou crie um analisador para cada thread. O lxml também permite a simultaneidade durante a validação (RelaxNG e XMLSchema) e a transformação XSL. Você pode compartilhar objetos RelaxNG, XMLSchema e XSLT entre threads.
Embora você também possa compartilhar analisadores entre encadeamentos, isso serializará o acesso a cada um deles, portanto, é melhor analisar os analisadores. copy () ou simplesmente usar o analisador padrão, se você não precisar de nenhuma configuração especial. O mesmo se aplica aos avaliadores XPath, que usam um bloqueio interno para proteger seus contextos de avaliação preparados. Portanto, é melhor usar instâncias de avaliador separadas nos encadeamentos.
Aviso: Antes do lxml 2.2, e especialmente antes do 2.1, havia vários problemas ao mover subárvores entre encadeamentos diferentes ou ao aplicar objetos XSLT de um encadeamento a árvores analisadas ou modificadas em outro. Se você precisar de código para rodar com versões mais antigas, geralmente deve evitar a modificação de árvores em outros tópicos além do que foi gerado. Embora isso deva funcionar em muitos casos, há certos cenários em que a terminação de um thread que analisa uma árvore pode falha o aplicativo se as subárvores dessa árvore forem movidas para outros documentos. Você deve estar no lado seguro ao passar as árvores entre as linhas, se você quer.
não modifique essas árvores e não mova seus elementos para outras árvores, ou não termine threads enquanto as árvores analisadas ainda estiverem em uso (por exemplo, usando um conjunto de encadeamentos de tamanho fixo ou encadeamentos de longa execução em cadeias de processamento)
Desde o lxml 2.2, até pipelines multi-thread são suportados. No entanto, observe que é mais eficiente fazer todo o trabalho em árvore dentro de um encadeamento do que permitir que vários encadeamentos funcionem em uma árvore, um após o outro. Isso ocorre porque as árvores herdam o estado do encadeamento que as criou, que deve ser mantido quando a árvore é modificada dentro de outro encadeamento.
Meu programa roda mais rápido se eu usar threads?
Depende. A melhor maneira de responder isso é o tempo e o perfil.
O bloqueio global de intérprete (GIL) no Python serializa o acesso ao interpretador, portanto, se a maior parte do seu processamento for feita em código Python (árvores caminhando, elementos modificadores, etc.), seu ganho será próximo de zero. Quanto mais o processamento de XML for movido para lxml, maior será o seu ganho. Se o seu aplicativo estiver vinculado à análise e serialização de XML, ou por expressões XPath muito seletivas e XSLTs complexos, sua aceleração em máquinas com vários processadores poderá ser substancial.
Veja a pergunta acima para saber quais operações liberam o GIL para suportar multi-threading.
Meu programa single-threaded funcionaria mais rápido se eu desligasse o thread?
Possivelmente sim. Você pode ver por si mesmo, compilando lxml inteiramente sem suporte a threads. Passe a opção --without-threading para setup. py ao criar lxml a partir da fonte. Você também pode construir libxml2 sem suporte a pthread (opção --without-pthreads), o que pode adicionar outro pouco de desempenho. Observe que isso deixará estruturas de dados internas totalmente sem proteção de encadeamento, portanto, verifique se você realmente não usa lxml fora do encadeamento do aplicativo principal nesse caso.
Por que não posso reutilizar as folhas de estilo XSLT em outros segmentos?
Desde versões mais recentes do lxml 2.0, você pode fazer isso. Há alguma sobrecarga envolvida, pois o documento resultante precisa de uma travessia de limpeza adicional quando o documento de entrada e / ou a folha de estilo foram criados em outros encadeamentos. No entanto, em uma máquina com vários processadores, o ganho de liberar a GIL cobre facilmente essa desvantagem.
Se você precisar mesmo do último desempenho, considere manter (uma cópia) da folha de estilo no armazenamento local do encadeamento e tente criar o (s) documento (s) de entrada no mesmo encadeamento. E não se esqueça de comparar seu código para ver se a maior complexidade do código realmente vale a pena.
Meu programa trava quando rodado com o mod_python / Pyro / Zope / Plone /.
Esses ambientes podem usar encadeamentos de uma maneira que pode não ser óbvia quando os encadeamentos são criados e o que acontece em qual encadeamento. Isso dificulta a garantia de que o suporte a threads do lxml seja usado de maneira confiável. Infelizmente, se surgirem problemas, eles são tão diversos quanto os aplicativos, por isso é difícil fornecer qualquer solução geralmente aplicável. Além disso, esses ambientes são tão complexos que os problemas se tornam difíceis de depurar e ainda mais difíceis de reproduzir de maneira previsível. Se você encontrar falhas em um desses sistemas, mas seu código é executado perfeitamente quando iniciado manualmente, o seguinte fornece algumas dicas para possíveis abordagens para resolver seu problema específico:
Certifique-se de usar versões recentes de libxml2, libxslt e lxml. Os desenvolvedores da libxml2 continuam corrigindo bugs em cada lançamento, e o lxml também tenta se tornar mais robusto contra possíveis armadilhas. Portanto, versões mais novas podem já corrigir seu problema de maneira confiável. A versão 2.2 do lxml contém muitas melhorias.
Verifique se as versões da biblioteca que você instalou são realmente usadas. Não confie no que o seu sistema operacional lhe diz! Imprima as constantes de versão em lxml. etree de dentro do seu ambiente de tempo de execução para certificar-se de que é o caso. Isso é especialmente um problema no MacOS-X quando novas versões de bibliotecas foram instaladas, além das bibliotecas do sistema desatualizadas. Por favor, leia a seção de bugs sobre o MacOS-X nesta FAQ.
se você usa o mod_python, tente configurar esta opção:
Houve uma discussão na lista de discussão sobre esse problema:
em um ambiente encadeado, tente inicialmente importar lxml. etree do encadeamento do aplicativo principal em vez de fazer importações pela primeira vez separadamente em cada encadeamento de trabalho gerado. Se você não puder controlar a geração de segmentos do seu servidor web / de aplicativos, uma importação de lxml. etree em sitecustomize. py ou usercustomize. py ainda poderá ser realizada.
compile o lxml sem o suporte ao encadeamento executando o setup. py com a opção --without-threading. Embora isso possa ser mais lento em determinados cenários em sistemas com vários processadores, ele também pode evitar que o aplicativo falhe, o que deve valer mais para você do que espiar o desempenho. Lembre-se de que o lxml é rápido de qualquer maneira, portanto a simultaneidade pode nem valer a pena.
procure por coisas extravagantes do XSLT, como o acesso a documentos estrangeiros ou passando por subárvores através de variáveis ​​XSLT. Isso pode ou não funcionar, dependendo do seu uso específico. Novamente, versões posteriores do lxml e libxslt fornecem suporte mais seguro aqui.
tente copiar árvores em locais suspeitos em seu código e trabalhe com elas em vez de uma árvore compartilhada entre threads. Observe que a cópia deve acontecer dentro do encadeamento de destino para ser eficaz, não no encadeamento que criou a árvore. Serializar em um thread e analisar em outro também é uma maneira simples (e rápida) de separar contextos de thread.
tente manter cópias locais de folhas de estilo XSLT, ou seja, uma por thread, em vez de compartilhar uma. Veja também a questão acima.
Você pode tentar serializar partes suspeitas do seu código com bloqueios de thread explícitos, desabilitando assim a simultaneidade do sistema de tempo de execução.
reporte-se na lista de discussão para ver se há outras maneiras de contornar seus problemas específicos. Não se esqueça de relatar os números das versões de lxml, libxml2 e libxslt que você está usando (veja a questão sobre como reportar um bug).
Observe que a maioria dessas opções prejudicará o desempenho e / ou a qualidade do seu código. Se você não tiver certeza do que fazer, por favor, pergunte na lista de discussão.
Análise e serialização.
Por que a opção pretty_print não reformata minha saída XML?
Impressão bonita (ou formatação) de um documento XML significa adicionar espaço em branco ao conteúdo. Essas modificações são inofensivas se impactarem apenas elementos no documento que não carregam dados (texto). Eles corrompem seus dados se eles impactarem elementos que contêm dados. Se lxml não puder distinguir entre espaços em branco e dados, isso não alterará seus dados. O espaço em branco é, portanto, adicionado apenas entre os nós que não contêm dados. Este é sempre o caso de árvores construídas elemento por elemento, portanto, nenhum problema deve ser esperado aqui. Para árvores analisadas, uma boa maneira de garantir que nenhum espaço em conflito seja deixado na árvore é a opção remove_blank_text:
Isso permitirá que o analisador solte os nós de texto em branco ao construir a árvore. Se você agora chamar uma função de serialização para imprimir bem essa árvore, o lxml pode adicionar novos espaços em branco à árvore XML para recuá-la.
Observe que a opção remove_blank_text também usa uma heurística se não tiver conhecimento definido sobre o espaço em branco ignorável do documento. Ele manterá os nós de texto em branco que aparecem após os nós de texto não em branco no mesmo nível. Isso evita que o XML do estilo de documento perca conteúdo.
O HTMLParser tem esse conhecimento estrutural integrado, o que significa que a maioria dos espaços em branco que aparecem entre as tags em documentos HTML não será removida por essa opção, exceto em locais em que é realmente ignorável, por exemplo, no cabeçalho da página, entre as tags da estrutura da tabela, etc. Portanto, também é seguro usar essa opção com o HTMLParser, pois ele manterá o conteúdo intacto a seguir (ou seja, não removerá o espaço que separa as duas palavras):
Se você quer ter certeza de que todo texto em branco é removido de um documento XML (ou apenas mais texto em branco do que o analisador por si só), você tem que usar um DTD para dizer ao analisador qual espaço em branco ele pode ignorar com segurança ou remover o espaços em branco ignoráveis ​​manualmente após a análise, por exemplo definindo todo o texto final para Nenhum:
Fredrik Lundh também tem uma função em nível Python para recuar XML anexando whitespace a tags. Ele pode ser encontrado em sua página de receita da biblioteca de elementos.
Por que o lxml não pode analisar meu XML de strings unicode?
Primeiro de tudo, o XML é explicitamente definido como um fluxo de bytes. Não é um texto Unicode. Dê uma olhada na especificação XML, é tudo sobre seqüências de bytes e como mapeá-las para texto e estrutura. Isso leva à regra número um: não decodifique seus dados XML por conta própria. Isso faz parte do trabalho de um analisador XML, e faz isso muito bem. Basta passar seus dados como um fluxo de bytes simples, ele sempre fará a coisa certa, por especificação.
Isso também inclui não abrir arquivos XML no modo de texto. Certifique-se de usar sempre o modo binário ou, melhor ainda, passe o caminho do arquivo para a função parse () do lxml para permitir que ele abra, leia e feche a si mesmo. Essa é a maneira mais simples e eficiente de fazer isso.
Dito isso, o lxml pode ler strings unicode do Python e até tentar suportá-las se a libxml2 não. Isso ocorre porque há um caso de uso válido para analisar XML de sequências de texto: fragmentos XML literais no código-fonte.
Entretanto, se a string unicode declarar uma codificação XML internamente (& lt;? Xml encoding = "."? & Gt;), a análise está fadada a falhar, já que essa codificação quase certamente não é a codificação real usada no Python unicode. O mesmo é verdadeiro para cadeias de caracteres unicode HTML que contêm meta tags charset, embora os problemas possam ser mais sutis aqui. O analisador HTML libxml2 pode não conseguir analisar as meta tags em HTML quebrado e pode acabar ignorando-as, portanto, mesmo se a análise for bem-sucedida, o tratamento posterior ainda poderá falhar com erros de codificação de caracteres. Therefore, parsing HTML from unicode strings is a much saner thing to do than parsing XML from unicode strings.
Note that Python uses different encodings for unicode on different platforms, so even specifying the real internal unicode encoding is not portable between Python interpreters. Não faça isso.
Python unicode strings with XML data that carry encoding information are broken. lxml will not parse them. You must provide parsable data in a valid encoding.
Can lxml parse from file objects opened in unicode/text mode?
Tecnicamente sim. However, you likely do not want to do that, because it is extremely inefficient. The text encoding that libxml2 uses internally is UTF-8, so parsing from a Unicode file means that Python first reads a chunk of data from the file, then decodes it into a new buffer, and then copies it into a new unicode string object, just to let libxml2 make yet another copy while encoding it down into UTF-8 in order to parse it. It's clear that this involves a lot more recoding and copying than when parsing straight from the bytes that the file contains.
If you really know the encoding better than the parser (e. g. when parsing HTML that lacks a content declaration), then instead of passing an encoding parameter into the file object when opening it, create a new instance of an XMLParser or HTMLParser and pass the encoding into its constructor. Afterwards, use that parser for parsing, e. g. by passing it into the etree. parse(file, parser) function. Remember to open the file in binary mode (mode="rb"), or, if possible, prefer passing the file path directly into parse() instead of an opened Python file object.
What is the difference between str(xslt(doc)) and xslt(doc).write() ?
The str() implementation of the XSLTResultTree class (a subclass of the ElementTree class) knows about the output method chosen in the stylesheet (xsl:output), write() doesn't. If you call write(), the result will be a normal XML tree serialization in the requested encoding. Calling this method may also fail for XSLT results that are not XML trees (e. g. string results).
If you call str(), it will return the serialized result as specified by the XSL transform. This correctly serializes string results to encoded Python strings and honours xsl:output options like indent . This almost certainly does what you want, so you should only use write() if you are sure that the XSLT result is an XML tree and you want to override the encoding and indentation options requested by the stylesheet.
Why can't I just delete parents or clear the root node in iterparse()?
The iterparse() implementation is based on the libxml2 parser. It requires the tree to be intact to finish parsing. If you delete or modify parents of the current node, chances are you modify the structure in a way that breaks the parser. Normally, this will result in a segfault. Please refer to the iterparse section of the lxml API documentation to find out what you can do and what you can't do.
How do I output null characters in XML text?
Don't. What you would produce is not well-formed XML. XML parsers will refuse to parse a document that contains null characters. The right way to embed binary data in XML is using a text encoding such as uuencode or base64.
Is lxml vulnerable to XML bombs?
This has nothing to do with lxml itself, only with the parser of libxml2. Since libxml2 version 2.7, the parser imposes hard security limits on input documents to prevent DoS attacks with forged input data. Since lxml 2.2.1, you can disable these limits with the huge_tree parser option if you need to parse really large, trusted documents. All lxml versions will leave these restrictions enabled by default.
Note that libxml2 versions of the 2.6 series do not restrict their parser and are therefore vulnerable to DoS attacks.
Note also that these "hard limits" may still be high enough to allow for excessive resource usage in a given use case. They are compile time modifiable, so building your own library versions will allow you to change the limits to your own needs. Also see the next question.
How do I use lxml safely as a web-service endpoint?
XML based web-service endpoints are generally subject to several types of attacks if they allow some kind of untrusted input. From the point of view of the underlying XML tool, the most obvious attacks try to send a relatively small amount of data that induces a comparatively large resource consumption on the receiver side.
First of all, make sure network access is not enabled for the XML parser that you use for parsing untrusted content and that it is not configured to load external DTDs. Otherwise, attackers can try to trick the parser into an attempt to load external resources that are overly slow or impossible to retrieve, thus wasting time and other valuable resources on your server such as socket connections. Note that you can register your own document loader in lxml, which allows for fine-grained control over any read access to resources.
Some of the most famous excessive content expansion attacks use XML entity references. Luckily, entity expansion is mostly useless for the data commonly sent through web services and can simply be disabled, which rules out several types of denial of service attacks at once. This also involves an attack that reads local files from the server, as XML entities can be defined to expand into their content. Consequently, version 1.2 of the SOAP standard explicitly disallows entity references in the XML stream.
To disable entity expansion, use an XML parser that is configured with the option resolve_entities=False . Then, after (or while) parsing the document, use root. iter(etree. Entity) to recursively search for entity references. If it contains any, reject the entire input document with a suitable error response. In lxml 3.x, you can also use the new DTD introspection API to apply your own restrictions on input documents.
Another attack to consider is compression bombs. If you allow compressed input into your web service, attackers can try to send well forged highly repetitive and thus very well compressing input that unpacks into a very large XML document in your server's main memory, potentially a thousand times larger than the compressed input data.
As a counter measure, either disable compressed input for your web server, at least for untrusted sources, or use incremental parsing with iterparse() instead of parsing the whole input document into memory in one shot. That allows you to enforce suitable limits on the input by applying semantic checks that detect and prevent an illegitimate use of your service. If possible, you can also use this to reduce the amount of data that you need to keep in memory while parsing the document, thus further reducing the possibility of an attacker to trick your system into excessive resource usage.
Finally, please be aware that XPath suffers from the same vulnerability as SQL when it comes to content injection. The obvious fix is to not build any XPath expressions via string formatting or concatenation when the parameters may come from untrusted sources, and instead use XPath variables, which safely expose their values to the evaluation engine.
The defusedxml package comes with an example setup and a wrapper API for lxml that applies certain counter measures internally.
XPath and Document Traversal.
What are the findall() and xpath() methods on Element(Tree)?
findall() is part of the original ElementTree API. It supports a simple subset of the XPath language, without predicates, conditions and other advanced features. It is very handy for finding specific tags in a tree. Another important difference is namespace handling, which uses the tagname notation. This is not supported by XPath. The findall, find and findtext methods are compatible with other ElementTree implementations and allow writing portable code that runs on ElementTree, cElementTree and lxml. etree.
xpath() , on the other hand, supports the complete power of the XPath language, including predicates, XPath functions and Python extension functions. The syntax is defined by the XPath specification. If you need the expressiveness and selectivity of XPath, the xpath() method, the XPath class and the XPathEvaluator are the best choice.
Why doesn't findall() support full XPath expressions?
It was decided that it is more important to keep compatibility with ElementTree to simplify code migration between the libraries. The main difference compared to XPath is the tagname notation used in findall() , which is not valid XPath.
ElementTree and lxml. etree use the same implementation, which assures 100% compatibility. Note that findall() is so fast in lxml that a native implementation would not bring any performance benefits.
How can I find out which namespace prefixes are used in a document?
You can traverse the document ( root. iter() ) and collect the prefix attributes from all Elements into a set. However, it is unlikely that you really want to do that. You do not need these prefixes, honestly. You only need the namespace URIs. All namespace comparisons use these, so feel free to make up your own prefixes when you use XPath expressions or extension functions.
The only place where you might consider specifying prefixes is the serialization of Elements that were created through the API. Here, you can specify a prefix mapping through the nsmap argument when creating the root Element. Its children will then inherit this prefix for serialization.
How can I specify a default namespace for XPath expressions?
Você não pode. In XPath, there is no such thing as a default namespace. Just use an arbitrary prefix and let the namespace dictionary of the XPath evaluators map it to your namespace. See also the question above.

Binary option chains

To configure an HTTPS server, the ssl parameter must be enabled on listening sockets in the server block, and the locations of the server certificate and private key files should be specified:
The server certificate is a public entity. It is sent to every client that connects to the server. The private key is a secure entity and should be stored in a file with restricted access, however, it must be readable by nginx’s master process. The private key may alternately be stored in the same file as the certificate:
in which case the file access rights should also be restricted. Although the certificate and the key are stored in one file, only the certificate is sent to a client.
The directives ssl_protocols and ssl_ciphers can be used to limit connections to include only the strong versions and ciphers of SSL/TLS. By default nginx uses “ ssl_protocols TLSv1 TLSv1.1 TLSv1.2 ” and “ ssl_ciphers HIGH:!aNULL:!MD5 ”, so configuring them explicitly is generally not needed. Note that default values of these directives were changed several times.
HTTPS server optimization.
SSL operations consume extra CPU resources. On multi-processor systems several worker processes should be run, no less than the number of available CPU cores. The most CPU-intensive operation is the SSL handshake. There are two ways to minimize the number of these operations per client: the first is by enabling keepalive connections to send several requests via one connection and the second is to reuse SSL session parameters to avoid SSL handshakes for parallel and subsequent connections. The sessions are stored in an SSL session cache shared between workers and configured by the ssl_session_cache directive. One megabyte of the cache contains about 4000 sessions. The default cache timeout is 5 minutes. It can be increased by using the ssl_session_timeout directive. Here is a sample configuration optimized for a multi-core system with 10 megabyte shared session cache:
SSL certificate chains.
Some browsers may complain about a certificate signed by a well-known certificate authority, while other browsers may accept the certificate without issues. This occurs because the issuing authority has signed the server certificate using an intermediate certificate that is not present in the certificate base of well-known trusted certificate authorities which is distributed with a particular browser. In this case the authority provides a bundle of chained certificates which should be concatenated to the signed server certificate. The server certificate must appear before the chained certificates in the combined file:
The resulting file should be used in the ssl_certificate directive:
If the server certificate and the bundle have been concatenated in the wrong order, nginx will fail to start and will display the error message:
because nginx has tried to use the private key with the bundle’s first certificate instead of the server certificate.
Browsers usually store intermediate certificates which they receive and which are signed by trusted authorities, so actively used browsers may already have the required intermediate certificates and may not complain about a certificate sent without a chained bundle. To ensure the server sends the complete certificate chain, the openssl command-line utility may be used, for example:
When testing configurations with SNI, it is important to specify the - servername option as openssl does not use SNI by default.
In this example the subject (“ s ”) of the GoDaddy server certificate #0 is signed by an issuer (“ i ”) which itself is the subject of the certificate #1, which is signed by an issuer which itself is the subject of the certificate #2, which signed by the well-known issuer ValiCert, Inc. whose certificate is stored in the browsers’ built-in certificate base (that lay in the house that Jack built).
If a certificate bundle has not been added, only the server certificate #0 will be shown.
A single HTTP/HTTPS server.
It is possible to configure a single server that handles both HTTP and HTTPS requests:
Prior to 0.7.14 SSL could not be enabled selectively for individual listening sockets, as shown above. SSL could only be enabled for the entire server using the ssl directive, making it impossible to set up a single HTTP/HTTPS server. The ssl parameter of the listen directive was added to solve this issue. The use of the ssl directive in modern versions is thus discouraged.
Name-based HTTPS servers.
A common issue arises when configuring two or more HTTPS servers listening on a single IP address:
With this configuration a browser receives the default server’s certificate, i. e. example regardless of the requested server name. This is caused by SSL protocol behaviour. The SSL connection is established before the browser sends an HTTP request and nginx does not know the name of the requested server. Therefore, it may only offer the default server’s certificate.
The oldest and most robust method to resolve the issue is to assign a separate IP address for every HTTPS server:
An SSL certificate with several names.
There are other ways that allow sharing a single IP address between several HTTPS servers. However, all of them have their drawbacks. One way is to use a certificate with several names in the SubjectAltName certificate field, for example, example and example . However, the SubjectAltName field length is limited.
Another way is to use a certificate with a wildcard name, for example, *.example . A wildcard certificate secures all subdomains of the specified domain, but only on one level. This certificate matches example , but does not match example and sub. example . These two methods can also be combined. A certificate may contain exact and wildcard names in the SubjectAltName field, for example, example and *.example .
It is better to place a certificate file with several names and its private key file at the http level of configuration to inherit their single memory copy in all servers:
Server Name Indication.
A more generic solution for running several HTTPS servers on a single IP address is TLS Server Name Indication extension (SNI, RFC 6066), which allows a browser to pass a requested server name during the SSL handshake and, therefore, the server will know which certificate it should use for the connection. SNI is currently supported by most modern browsers, though may not be used by some old or special clients.
Only domain names can be passed in SNI, however some browsers may erroneously pass an IP address of the server as its name if a request includes literal IP address. One should not rely on this.
In order to use SNI in nginx, it must be supported in both the OpenSSL library with which the nginx binary has been built as well as the library to which it is being dynamically linked at run time. OpenSSL supports SNI since 0.9.8f version if it was built with config option “--enable-tlsext”. Since OpenSSL 0.9.8j this option is enabled by default. If nginx was built with SNI support, then nginx will show this when run with the “-V” switch:
However, if the SNI-enabled nginx is linked dynamically to an OpenSSL library without SNI support, nginx displays the warning:
Compatibilidade.
The SNI support status has been shown by the “-V” switch since 0.8.21 and 0.7.62. The ssl parameter of the listen directive has been supported since 0.7.14. Prior to 0.8.21 it could only be specified along with the default parameter. SNI has been supported since 0.5.23. The shared SSL session cache has been supported since 0.5.6.
Version 1.9.1 and later: the default SSL protocols are TLSv1, TLSv1.1, and TLSv1.2 (if supported by the OpenSSL library). Version 0.7.65, 0.8.19 and later: the default SSL protocols are SSLv3, TLSv1, TLSv1.1, and TLSv1.2 (if supported by the OpenSSL library). Version 0.7.64, 0.8.18 and earlier: the default SSL protocols are SSLv2, SSLv3, and TLSv1.
Version 1.0.5 and later: the default SSL ciphers are “ HIGH:!aNULL:!MD5 ”. Version 0.7.65, 0.8.20 and later: the default SSL ciphers are “ HIGH:!ADH:!MD5 ”. Version 0.8.19: the default SSL ciphers are “ ALL:!ADH:RC4+RSA:+HIGH:+MEDIUM ”. Version 0.7.64, 0.8.18 and earlier: the default SSL ciphers are.

Binary option chains

Serialization is the process of converting the state of an object into a form that can be persisted in a storage medium or transported across processes/machines. The opposite of serialization is deserialization which is a process that converts the outcome of serialization into the original object. The Framework offers two serialization technologies, binary and XML.
Binary Serialization.
Binary serialization preserves type system fidelity. Type system fidelity denotes that type information is not lost in the serialization process. For example when serialization some class say MyClass , type fidelity ensures that during deserialization, an object of type MyClass will be constructed. Note that to leverage type system fidelity, both ends participating in serialization and deserialization must use the same type system. Type fidelity and hence binary serialization is useful when you want to pass objects between clients and servers. Note that XML Serialization does not provide type fidelity.
During the serialization process, public and private fields of the object and the name of the class and the containing assembly are converted to a stream of bytes written to a data stream. During deserialization, an exact replica of the object along with type information is reconstructed.
XML Serialization.
Serialization Concepts.
Why would you want to use serialization. The ability to convert objects to and from a byte stream can be incredibly useful. Aqui estão alguns exemplos:
An application's state (i. e., object graph) can be saved in a disk file or database table ( Persistent Storage) , and then restored the next time the application runs. In fact, ASP saves and restores session state using serialization and de-serialization. A set of object can be easily copied into the clipboard and then pasted into some other application. In fact, Windows Forms uses this procedures when you copy (serialize) and paste (deserialize) GUI elements. An object can be cloned using serialization and set aside while the user manipulates the original object. A set of objects can be easily sent over the network to a process running in another machine ( Marshal By Value) . In fact, Remoting using serialization and deserialization for sending/receiving marshal-by-value objects. Once object have been serialized, it is quite easy to perform many other functions such as compression, encryption, and so on.
Persistent Storage.
Imagine having an application with 100s of objects, and you are required to save the state (fields) of each object to a database. Serialization provides a convenient design pattern for achieving this objective with minimal effort.
The CLR manages how objects are laid out in memory and provides an automated serialization mechanism using reflection. When an object is serialized, the name of the class, its containing assembly and all fields are written to storage. When an object is serialized, the CLR also keeps track of all referenced objects already serialized to ensure that the same object is not serialized more than once. The serialization architecture provided with the Framework handles object graphs and circular references automatically. The only requirement for objects referenced from the serialized object is that those referenced objects be marked as [Serializable] . Else, the CLR will throw an exception when the serialize attempts to serialize an un-serializable object.
Marshal By Value.
Objects are only valid in the application domain where they were created. Attempting to pass an object as a parameter or as a return value across application domains will fail unless:
The object derives from MarshalByRefObject , or The object is marked with [Serializable] .
When an object derives form MarshalByRefObject , an object reference will be passed from one application domain to another rather than the object itself. When an object is marked with [Serializable] , the object will be automatically serialized, transported from one application domain to another and then deserialized to produce an exact copy of the object in the second application domain. Note then that while MarshalByRefObject passes a reference, [Serializable] causes the object to be copied .
Serialization Guidelines.
When designing new classes, serialization should be one of the issues to consider. Por exemplo,
Will this class be sent across application domains? Will this class ever be used with remoting? Will this class ever need to be persisted? Will this class be used as a base class to create derived classes that need to be serialized?
When in doubt, mark the class as serializable unless,
The class will never have to cross application domains. The class stored references that are only applicable to the current instance. Some of the fields contain sensitive information.
Basic Serialization.
The class that performs the actual serialization and deserialization is called a formatter . A formatter is a class type that implements System. Runtime. Serialization. IFormatter interface to serialize/deserialize an object graph (a graph is a generalized tree, and in a serialization/deserialization context, an object graph represents a generalized tree with the serialized/deserialized object being at the root of the tree. The formatter walks the object graph to make sure that all objects are serialized/deserialized)
When a type is designed, the developer must make an explicit decision as to whether to allow instances of this type of be serialized or not. Observe os seguintes pontos:
By default types are not serializable . The easiest way to make a class serializable is to mark it with [Serializable] attribute. The [Serializable] attribute can be applied to reference types, value types, enumerators and delegates only . The [Serializable] attribute is not inherited and derived classes must specify the [Serializable] attribute if they are intended to be serialized. If a derived class is marked with [Serializable] , all base classes in the class hierarchy must also be marked with [Serializable] . If one of the base classes is not marked with [Serializable] , you will get the following error:
You will also get the same exact error if a class is marked with [Serializable] and one or more of its field members are serializable. This situation happens when the field member is an object of a class not marked with [Serializable] :
The following code provides an example on how to perform basic serialization / deserialization:
public class MyBasicClass.
private string strPrivate;
public string strPublic;
private void btnBasicBinary_Click(object sender, System. EventArgs e)
private void SerializeMyBasicCObject()
// Create a serializable instance.
MyBasicClass ob = new MyBasicClass( "Private field data", "Public field data" );
// derived from System. IO. Stream abstract base class - i. e., you can serialize to MemoryStream,
// FileStream, NetworkStream and so on)
Stream stream = new FileStream( "TestFile. bin", FileMode. Create, FileAccess. Write, FileShare. Write);
System. Runtime. Serialization. IFormatter formatter = new System. Runtime. Serialization. Formatters. Binary. BinaryFormatter();
formatter. Serialize( stream, ob );
private void DeSerializeMyBasicCObject()
// Read the file back into a stream.
Stream stream = new FileStream( "TestFile. bin", FileMode. Open, FileAccess. Read, FileShare. Read);
System. Runtime. Serialization. IFormatter formatter = new System. Runtime. Serialization. Formatters. Binary. BinaryFormatter();
MyBasicClass ob = (MyBasicClass)formatter. Deserialize( stream );
Contents of TestFile. bin which holds a serialized object of type MyBasicClass (relevant parts have been highlighted):
яяяя
 KSerialization, Version=1.0.1614.14171, Culture=neutral, PublicKeyToken=null
Serialization. MyBasicClass
strPrivate strPublic

 Private field data Public field data
Note the following:
The private member strName in class MyBasicClass has been serialized. In this aspect, binary serialization differs from XML serialization which serializes only public fields . It is important to note that constructors are not called when objects are deserialized . If MyBasicClass class derived from a base class that was not marked with [Serializable] a SerializationException will be generated.
All objects serialized with BinaryFormatter can be deserialized with it, which makes it the ideal choice for serializing/deserializing objects on the platform. If portability is a requirement, use SoapFormatter instead. You can use the same code above except that BinaryFormatter should be replaced with SoapFormatter as follows:
System. Runtime. Serialization. IFormatter formatter = new System. Runtime. Serialization. Formatters. Binary. BinaryFormatter() ;
System. Runtime. Serialization. IFormatter formatter = new System. Runtime. Serialization. Formatters. Soap. SoapFormatter() ;
The SoapFormatter produces the following serialized object when used in the code above;
<SOAP-ENV:Envelope xmlns:xsi="w3/2001/XMLSchema-instance" xmlns:xsd="w3/2001/XMLSchema" xmlns:SOAP-ENC="schemas. xmlsoap/soap/encoding/" xmlns:SOAP-ENV="schemas. xmlsoap/soap/envelope/" xmlns:clr="schemas. microsoft/soap/encoding/clr/1.0" SOAP-ENV:encodingStyle="schemas. xmlsoap/soap/encoding/">
<a1:MyBasicClass id="ref-1" xmlns:a1="schemas. microsoft/clr/nsassem/Serialization/Serialization%2C%20Version%3D1.0.1614.15098%2C%20Culture%3Dneutral%2C%20PublicKeyToken%3Dnull">
<strPrivate id="ref-3">Private field data</strPrivate>
<strPublic id="ref-4">Public field data</strPublic>
A serialized object graph can be sent to many locations: same process, different process on the same machine, different process on a different machine, and so on. In certain cases, an object might want to know where it will be deserialized so that it initialize its state differently. For example, an object that wraps a Semaphore handle might decide to serialize its handle if it knows that deserialization will take place in the same process. The object might decide to serialize the semaphore's name if it knows that deserialization will take place in the same machine. Finally, the object might decide to throw an exception if it knows that deserialization will take place in a different machine. In other words, how does the object know where it is being deserialized?
The answer is simple: StreamingContext . A method that receives a StreamingContext object can use StreamingContext. State property to determine the source or destination of the object being serialized/deserialized. The following example shows how serialization can be used to clone an object:
private object DeepClone( object obSource )
// Initialize a storage medium to hold the serialized object.
Stream stream = new MemoryStream();
BinaryFormatter formatter = new BinaryFormatter();
formatter. Context = new StreamingContext( StreamingContextStates. Clone );
formatter. Serialize( stream, obSource );
object obTarget = formatter. Deserialize( stream );
It is possible (and can be quite useful) to serialize multiple objects into the same stream:
private Stream SerializeMultipleObjects()
// Initialize a storage medium to hold the serialized object.
Stream stream = new MemoryStream();
BinaryFormatter formatter = new BinaryFormatter();
formatter. Serialize( stream, obOrders );
formatter. Serialize( stream, obProducts );
formatter. Serialize( stream, obCustomers );
private void DeSerializeMultipleObject(Stream stream)
// Construct a binary formatter.
BinaryFormatter formatter = new BinaryFormatter();
Orders obOrders = (Orders)formatter. Deserialize( stream );
Products obProducts = (Products)formatter. Deserialize( stream );
Customers obCustomers = (Customer)formatter, Deserialize( stream )
Formatters.
Formatters know how to serialize the complete object graph by walking the object graph and referring to each object's metadata. The Serialize method uses reflection to discover instance fields in each object's type, and if any of these fields refer to other objects, the Serialize method will know how to serialize them as well.
Formatters have very intelligent algorithms. They know how to serialize each object in the graph to the stream no more than once. For example, if two objects in the object graph are actually the same object (they both reference the same location), then the formatter detects this and serializes the object only once (and hence avoids entering into an infinite loop). As for deserialization, the formatter examines the contents of the byte stream, constructs objects of all instances that are in the stream ( recall that constructors are not called when objects are deserialized ), and initializes fields in all these objects so they have the same values they had when the object graph was serialized.
Fromatters and Assembly Names.
Serializing.
When formatters serialize an object, the full name of the type (i. e., MyNamespace. MySubNamespace. MyClass ) and the name of the assembly containing the type (i. e., MyAssembly. dll ) are written to the byte stream. By default, BinaryFormatter and SoapFormatter types output the assembly's full identity (name, version, public key, and culture). To make these formatter output just the simple assembly name for each serialized type, set the formatter's AssemblyFormat property to FormatterAssemblyStyle. Simple (the default is FormatterAssemblyStyle. Full .)
Deserializing.
When formatters deserialize an object, they first get the assembly identity and ensure that the assembly is loaded into the executing AppDomain . How this assembly is loaded depends on the value of the formatter's AssemblyFormat :
The assembly is loaded using System. Reflection. Assembly. Load which first looks in the GAC and then looks in the application's directory. If the assembly is not found, an exception is thrown and deserialization fails.
The assembly is loaded using System. Reflection. Assembly. LoadWithPartialName which first looks in the application's directory, and if not found, looks in the GAC for the highest version numbered assembly with the same file name.
After an assembly has been loaded, the formatter looks in the assembly for a type matching that of the object being serialized. If there is no matching type, an exception is thrown and no more objects can be serialized. If a matching type is found, an instance type is created (constructor not called) and its fields are initialized from the values present in the stream.
Formatters and dynamically loaded assemblies.
Some application may use Assembly. LoadFrom to load an assembly and then construct objects from types defined in the assembly. Such object can be serialized with no problems. However, when deserializing such objects, the formatter attempts to load the assembly using Assembly. Load or Assembly. LoadWithPartialName instead of calling Assembly. LoadFrom method. In most cases, the assembly will fail to load (most likely not found) and an exception will be thrown.
The solution is to implement a method whose signature matches the System. ResolveEventHandler delegate and register this method with System. AppDomain. AssemblyResolve event just before deserializing an object (unregister this method with the event after deserializing the object). Whenever the formatter fails to load an assembly, the CLR will call your System. ResolveEventHandler method passing in the identity of the assembly that failed to load. This information can be used by the method to extract the assembly file name and construct the path where the application knows the assembly file can be found. This method then calls Assembly. LoadFrom to load the assembly and return the resulting assembly reference back from the System. ResolveEventHandler method.
Formatters and internal processing.
This section provides more insight into how a formatter serializes/deserializes an object. This knowledge will help you understand how this process really works.
The following steps describe how a formatter automatically serializes an object whose type has the [Serializable] attribute applied to it:
The formatter calls FormatterServices. GetSerializableMemebers method ( System. Runtime. Serialization namespace) to return an array of MemberInfo objects, one for each serializable instance field (these MemberInfo object are obtained via serialization). The object being serializes and the MemberInfo array is passed to FormatterServices. GetObjectData method ( System. Runtime. Serialization namespace) to return an array objects, where each object in the array identifies a field being serialized. This object array and the MemberInfo array passed to GetObjectData are parallel - element 0 in the object array is the value of the member identified by element 0 in the MemberInfo array. The formatter writes the assembly's identity and the type's full name to the stream. Finally, the formatter enumerates over the elements in the two arrays, writing each member's name and value to the stream.
The following steps describe how a formatter automatically deserializes an object whose type has the [Serializable] attribute applied to it:
The formatter calls FormatterServices. GetUninitializedObject method ( System. Runtime. Serialization namespace) to allocate memory for a new object, but does not call the constructor. The formatter constructs and initializes a MemberInfo array just like it did before by calling FormatterServices. GetSerializableMemebers method This method returns the set of fields that were serialized and that now need to be deserialized. The formatter creates and initializes an object array from the data contained in the byte stream. The reference to the newly allocated object, the MemberInfo array, and the paralled object array (containing filed values) is passed to FormatterServices. PopulateObectMembers method.
Selective Serialization.
In general it is recommended that most types be made serializable since this grants a lot of flexibility to your users. However, recall that serialization reads all of an object's fields regardless of whether the fields are declared as public, internal, protected, or private. You might want to use selective serialization if the type contains sensitive data or if the data had no meaning if transferred (i. e., windows handles, thread ids, etc. )
Selective serialization means controlling which fields should serialize and which fields should not serialize. For example, it does not make sense to serialize a field containing the thread ID of the currently executing thread because that thread will probably not be alive when the object is deserialized. Selective serialization is achieved by annotating the class with [Serializable] attribute and then further annotating fields that should not be serialized with [NonSerialized] :
public class Circle.
// The following fields will be serialized.
private double dRadius;
[NonSerialized] private double dArea;
dArea = Math. PI * dRadius * dRadius;
// Construct a Circle object.
Circle cir = new Circle( 10.0 );
When an object is constructed, the radius and the area are set to proper values. When the object gets serialized, only the value of dRadius will be written to the stream. This is exactly what you want since dArea is a calculated field and there is no need to serialize it. However, you have a problem when the object is deserialized - when deserialized, the Circle object will get its dRadius value set to 10.00, but dArea will be set to 0! The following code shows how to fix the problem:
public class Circle : IDeserializationCallback.
// The following fields will be serialized.
private double dRadius;
[NonSerialized] private double dArea;
dArea = Math. PI * dRadius * dRadius;
public void IDeserializationCallback. OnDeserialization ( object oSender )
dArea = Math. PI * dRadius * dRadius;
Objects are reconstructed from the inside out, and calling methods during deserialization can have unwanted effects since the methods called might refer to objects that have not be deserialized yet. During deserialization, formatters check if the type implements IDeserializationCallback interface, and if so, add this object to an internal list. After all objects have been deserialized, the formatter walks this list and calls each object's OnDeserialization . OnDeserialization therefore, should be used to do any additional work that would be necessary to fully deserialize the object. A hashtable is a typical example of a class that is difficult to deserialize without using the IDeserialiationCallback interface.
Custom Serialization.
Introdução.
The [Serializable] attribute is used to indicate that the given class can be serialized. The [Serializable] attribute further indicates that the class will be serialized using the default serialization process which excludes all fields marked with the [NonSerialized] attribute. ISerializable interface on the other hand allows an object to completely control its own serialization and deserialization processes. In other words, apply [Serializable] attribute to indicate that the class can be serialized and implement ISerializable interface to control the serialization process. Note that if ISerializable is implemented, then [Serializable] must also be applied, otherwise, the following error is generated:
Controlling the serialization process through ISerializable involves implementing the GetObjectData() and a special constructor:
GetObjectData is invoked when the object is serialized. The special constructor is invoked when the object is deserialized.
The code below shows how to implement ISerializable :
// A simple serializabel class that implements its serialization/deserialization processes.
public class MySerializableClass : System. Runtime. Serialization. ISerializable.
private string strPrivate = String. Empty;
public string strPublic = String. Empty;
public SomeClass obMyClass = null; // SomeClass must be marked [Serializable]
public MySerializableClass( string s1, string s2 )
obMyClass = new SomeClass( 999 );
// this required constructor. Note that no warning will be given if this constructor is absent and.
// an exception will be thrown if you attempt to deserialize this object.
// It is a good idea to make the constructor protected unless the class was sealed, in which case.
// the constructor should be marked as private. Making the constructor protected ensures that users.
// cannot explicitly call this constructor while still allowing derived classes to call it. The same.
// logic applies if this class was sealed except that there will be no derived classes.
protected MySerializableClass( SerializationInfo info, StreamingContext ctx )
strPrivate = info. GetString( "s1" );
strPublic = info. GetString( "s2" );
obMyClass = (SomeClass) info. GetValue( "ob", typeof(SomeClass) ); // deserializing an object.
// as name/value pairs to SerializationInfo object. Here you decide which member variables to serialize.
// Note that any name can be used to identify each serialized field.
public virtual void GetObjectData( SerializationInfo info, StreamingContext ctx )
info. AddValue( "s1", strPrivate );
info. AddValue( "s2", strPublic );
into. AddValue( "ob", obMyClass ); // Serializing an object.
private void btnSerializable_Click(object sender, System. EventArgs e)
/* Prepare for serialization */
MySerializableClass ob = new MySerializableClass( "hello", "world");
System. Runtime. Serialization. IFormatter formatter = new System. Runtime. Serialization. Formatters. Binary. BinaryFormatter();
Stream stream = new FileStream( "TestFile2.bin", FileMode. Create, FileAccess. Write, FileShare. Write);
formatter. Serialize( stream, ob );
System. Runtime. Serialization. IFormatter formatter2 = new System. Runtime. Serialization. Formatters. Binary. BinaryFormatter();
Stream stream2 = new FileStream( "TestFile2.bin", FileMode. Open, FileAccess. Read, FileShare. Read);
MySerializableClass ob2 = (MySerializableClass)formatter2.Deserialize( stream2 );
When a formatter serializes an object graph, it looks at each object's type. If the type implements the ISerializable interface, the formatter will construct a new SerializationInfo object which contains the actual set of values that should be serialized for the object. Once the SerializationInfo object is constructed, the formatter calls the type's GetObjectData passing it the SerializationInfo object. The GetObjectData method is responsible for determining the state of the object that should be serialized, and adds this information to the SerializationInfo object. GetObjectData indicates which information should be serializes by calling one of the overloaded AddValues methods. AddValue should be called once for each piece of data that you want to serialize.
Always calls AddValue to add serialization information for your type . If a field's type ( SomeClass in the example) above implements ISerializeble , do not call that object's GetObjectData method. Instead, call AddValue to add the object. The formatter will determine that the object implements ISerializable and it will call GetObjectData for you.
As for deserialization, the formatter extracts an object from the stream and allocates it memory (by calling FormatterServices. GetUninitializedObject ). Initially, all of this object's fields are set to null and/or 0. Then the formatter checks if the type implements ISerializable , and if so, the formatter calls a special constructor whose signature is similar to tat GetObjectData . After the formatter finishes executing this constructor, the object should be fully deserialized. Because the deserialized type may include fields that refer to other objects, you should not execute any code in the special constructor that can access members on these objects as they are not yet fully constructed.
If the deserialized type must call members on referenced objects, then consider implementing IDeserializationCallback . This concept was discussed here.
You are free to derive a class that inherits from a base class that implements the ISerializable interface. If the derived class does not implement ISerializable (i. e., has no special serialization/deserialization requirements), you do not need to do anything special (you still do not need to do anything even if the derived class was marked [Serializable] ). However, if the derived class implements the ISerializable interface as well, then you need to observe the following:
The derived class must implement both GetObjectData and the special constructor. GetObjectData implementation must first call the base class's GetObjectData before calling any AddValue methods. The special constructor must call the base class's special constructor.
These requirements are illustrated below:
public class MyDerivedSerializableClass : MySerializableClass.
public int nNum;
public MyDerivedSerializableClass( string s1, string s2 ) : base( s1, s2 ) <>
nNum = info. GetInt32( "n1" );
public override void GetObjectData( SerializationInfo info, StreamingContext ctx )
base. GetObjectData( info, ctx );
info. AddValue( "n1", nNum );
private void btnDerivedSerializable_Click(object sender, System. EventArgs e)
/* Prepare for serialization */
MyDerivedSerializableClass ob = new MyDerivedSerializableClass("hello", "world");
System. Runtime. Serialization. IFormatter formatter = new System. Runtime. Serialization. Formatters. Binary. BinaryFormatter();
Stream stream = new FileStream( "TestFile3.bin", FileMode. Create, FileAccess. Write, FileShare. Write);
formatter. Serialize( stream, ob );
System. Runtime. Serialization. IFormatter formatter2 = new System. Runtime. Serialization. Formatters. Binary. BinaryFormatter();
Stream stream2 = new FileStream( "TestFile3.bin", FileMode. Open, FileAccess. Read, FileShare. Read);
MyDerivedSerializableClass ob2 = (MyDerivedSerializableClass)formatter2.Deserialize( stream2 );
This example shows the general approach taken to serialize/deserialize the System. Collections. Hashtable class. Note the use of IDeserializationCallback to populate the hashtable's keys and values; IDeserializationCallback will only be called after all objects in the Hashtable class (including the actual keys and values) have been completely deserialized:
public class Hashtable : ISerializable, IDeserializationCallback.
//Internal data members not shown.
SerializationInfo siSaved = null;
protected Hashtable( SerializationInfo si, StreamingContext ctxt )
// Save the serialization info. We want to postpone populating the.
// hashtable until the key/value object have been deserialized.
public virtual void GetObjectData( SerializationInfo si, StreamingContext ctxt )
// Call AddValue methods on SerializationInfo object to manually.
// add most fields of this Hashtable object (fields and AddValue.
// Create object arrays to hold the Hashtable's keys and values.
object[] keys = new object[ nHashtableEntryCount ];
object[] values = new object[ nHashtableEntryCount ];
// Hashtable (not shown)
si. AddValue( "key", keys );
si. AddValue( "values", values );
public virtual void OnDeserialization( object oSender )
// All objects in the stream have been deserialized. We can populate.
// the Hashtable object now.
object[] keys = (object[])siSaved. GetValue( "keys", typeof(object[]) );
object[] values = (object[])siSaved. GetValue( "values", typeof(object[]) );
for (int x = 0; x < keys. Length; x++ )
Add( keys[x], values[x] );
ISerializable and IFormatConverter.
When the SerializationInfo object is being constructed by the formatter, it ( SerializationInfo) is passed an object whose type implements IFormatterConverter interface. This object knows how to convert values between the core types, such as converting an Int32 to Int64 . To convert a value between other non-core types, this object casts the serialized (or original value) to an IConvertible interface and then calls the appropriate interface method.
To allows objects of a serializable type to be serialized as different types, consider having this type implement the IConvertible interface. The IFormatterConverter is only used when deserializing objects and when calling a Get method whose type does not match the type of the value in the stream.
ISerializable and Base types.
ISerializable allows you to take full control over how a type gets serialized/deserialized. This power comes at a cost since the type is now responsible for serializing all its base type's fields as well. Recall the following limitations:
If a derived class is marked with [Serializable] , all base classes in the class hierarchy must also be marked with [Serializable] . If one of the base classes is not marked with [Serializable] , you will get the following error:
public class Person.
public class Employee : Person.
public class Manager : Employee, ISerializable.
Class Manager wants to take control of its own serialization by implementing ISerializable , but its base classes do not implement ISerializable . Class Manager cannot therefore, call the base's implementation of GetObjectData and the special constructor because simply they do not exist. Class Manager must perform the base class serialization manually. In the code below, FormatterServices provides static methods to aid with the serialization process The full code is shown below:
public class Person.
private string strTitle;
public Person(string s)
public class Employee : Person.
private string strTitle;
public Employee(string s) : base(s)
public class Manager : Employee, ISerializable.
private string strTitle;
public Manager( string s ) : base(s)
public virtual void GetObjectData( SerializationInfo si, StreamingContext ctxt )
// Serialize values from this class.
si. AddValue( "title", strTitle );
System. Reflection. MemberInfo[] aMI = FormatterServices. GetSerializableMembers ( this. GetType(), ctxt );
for (int i = 0; i < aMI. Length; i++)
// Do not serialize fields from this base class. They have already been serialized.
// in the AddValue method above.
if (aMI[i].DeclaringType == this. GetType() ) continue;
si. AddValue( aMI[i].Name, ((FieldInfo)aMI[i]).GetValue( this ) );
protected Manager( SerializationInfo si, StreamingContext ctxt) : base( String. Empty )
// Note that call above to the base class constructor that takes a string. If base.
// class implemented ISerializable, we would have called the base's special.
System. Reflection. MemberInfo[] aMI = FormatterServices. GetSerializableMembers ( this. GetType(), ctxt );
for (int i = 0; i < aMI. Length; i++)
// Do not de-serialize fields from this base class. They have already been serialized.
// in the AddValue method above.
if (aMI[i].DeclaringType == this. GetType() ) continue;
object oValue = si. GetValue( aMI[i].Name, ((FieldInfo)aMI[i]).FieldType ); // Note the casting!
((FieldInfo)aMI[i]).SetValue( this, oValue );
strTitle = si. GetString( "title" );
Manager man = new Manager( "Yazan" );
MemoryStream ms = new MemoryStream();
BinaryFormatter bf = new BinaryFormatter();
bf. Serialize( ms, man );
ms. Seek( 0, SeekOrigin. Begin );
Manager man2 = (Manager)bf. Deserialize( ms );
ISerializable and Proxies.
This section shows how to design a type that can serialize/deserialize itself to a different type. has many examples of this - for example, in Remoting when a client creates an instance of an SAO, the client gets a proxy object. While the type of the proxy is transparent to the client code (i. e., the client thinks it has an actual instance of an SAO), the type of the proxy object is different than the type of the server object. The following example illustrates how a serialized type is deserialized into another type:
public sealed class MyClass : ISerializable.
private string strName = "MyClass";
public MyClass( string s )
// The special constructor will never be called becuase we will not be.
// deserializing instances of this class. Therefore, special constructor not provided.
public void GetObjectData(SerializationInfo si, StreamingContext ctxt)
si. AddValue( "name", strName );
// No other values need to be added.
public sealed class MyClassProxy ISerializable.
private string strName = "MyClassProxy";
public MyClassProxy(SerializationInfo si, StreamingContext ctxt)
// public void GetObjectData(SerializationInfo si, StreamingContext ctxt) <>
MyClass ob = new MyClass( "Yazan" );
MemoryStream stream = new MemoryStream();
BinaryFormatter bf = new BinaryFormatter();
bf. Serialize( stream, ob );
stream. Seek( 0, SeekOrigin. Begin );
MyClassProxy obProxy = (MyClassProxy)bf. Deserialize( stream );
Note that the proxy type must derive from I S erializable , otherwise, the following error is generated.
Serialization Steps.
Object serialization proceeds as follows when Serialize is called on a formatter object:
Versão.
Because serialization deals with member variables and not interfaces, pay special attention to adding / removing variables from classes that will be serialized across versions. This is especially true for classes that do not implement ISerializable as any addition / deletion / type modification of member field means that existing objects of the same type cannot be deserialized if they were serialized with a previous version.
If the state of the object may have to change between versions, then you have two choices:
Implement ISerializable to take precise control over serialization and deserialization. Here you have total control over what gets serialized and deserialized. Mark non-essential member variables with [NonSerialized] . This option should by used only when you expect minor changes between versions.
Advanced Serialization.
Surrogates.
The discussion so far concentrated on how to modify a type's implementation so that the type can control its serialization and deserialization. The Framework also allows code that is not part of a type's implementation to override how a type serializes/deserializes itself. In other words, class A (referred to as the surrogate ) can control how class B gets serialized/deserialized. There are two main reasons why application code might want to override a type's serialization/deserialization behavior:
Provides the ability to serialize a type that was not originally designed to be serialized. Provides a way to map one version of a type to another version of another type.
The basic mechanism is as follows: Define a surrogate type that takes over the actions required to serialize/deserialize an existing type by implementing ISerializationSurrogate . Then you register an instance of the surrogate type with the formatter, telling the formatter which existing type your surrogate type is responsible for acting upon. When the formatter detects that it is trying to serialize/deserialize an instance of the exiting type, it will call methods defined by the surrogate type. The following example illustrates:
// This class is not marked as [Serializable]. However, we can still serialize it using a serialization surrogate.
public class Programmer.
public string strName = String. Empty;
public DateTime dtDOB;
// from ISerializationSurrogate). During serialization/deserialization, it will get an instance of the actual object.
// (a Programmer instance) which will be used to obtain/set all required state.
public class ProgrammerSurrogate : ISerializationSurrogate.
public void GetObjectData(object obj, SerializationInfo info, StreamingContext context)
// Get the real object.
Programmer p = (Programmer)obj;
// been present otherwise if Programmer type was serializable.
info. AddValue( "name", p. strName );
info. AddValue( "dob", p. dtDOB );
// to get the value of those fields.
public object SetObjectData(object obj, SerializationInfo info, StreamingContext context, ISurrogateSelector selector)
// Get the real object. The object's fields are all null and no constructor has been called.
Programmer pOb = (Programmer)obj;
pOb. strName = info. GetString( "name" );
pOb. dtDOB = info. GetDateTime( "dob" );
public void TestSurrogates()
// Select the serialization surrogate to delegate the serialization/deserialization process to.
// Note that the SurrogateSelector type is used to indicate the type that will act as a substitute for the actual object.
SurrogateSelector ss = new System. Runtime. Serialization. SurrogateSelector();
ss. AddSurrogate( typeof(Programmer), // The type for which a surrogate is required.
new StreamingContext( StreamingContextStates. All), // Context-specific data.
new ProgrammerSurrogate() ); // The surrogate to call for type indicated in the 1st parameter.
BinaryFormatter bf = new BinaryFormatter();
Programmer p = new Programmer("yazan", Convert. ToDateTime( "1/1/1990") );
MemoryStream ms = new MemoryStream();
bf. Serialize( ms, p ); // Calls GetObjectData for the specified SurrogateSelector object passing p as the first parameter.
Programmer p2 = (Programmer)bf. Deserialize( ms );
The following window shows the state of the serialized object p and the deserialized object p2:
Note in the preceding code that the surrogate type ProgrammerSurrogate does not have intimate knowledge of the Programmer type. And while the surrogate type ( ProgrammerSurrogate ) can easily access the fields of the Programmer type, it cannot do the same with its non-public fields. With non-public fields, reflection would have to be used (subject to security)
Note that SurrogateSelector type maintain an internal hashtable. When SurrogateSelector. AddSurrogate is called, the Type and the StreamingContext make up the key and the ISerializeSurrogate object makes up the key's value. If a key with the same Type and SurrogateSelector already exits, an exception is thrown. By specifying the same type but with a different StreamingContext object, you can register different surrogate types that know how to serialize and deserialize in different ways.
Surrogate Selector Chains.
Multiple SurrogateSelector objects can be chained together. For example, one SurrogateSelector can be used to maintain a set of serialization surrogates for serializing types into proxies that can be remoted, while another SurrogateSelector can be used to convert version 1 types into version 2 types. When you have more than one SurrogateSelector objects, you must chain them into a linked list. ISurrogateSelector interface, which is implemented by SurrogateSelector types, implements the appropriate methods for chaining.
Overriding Assembly / Type information.
Recall the following basic points:
When serializing an object, the formatter outputs the type's full name, as well as the full name of the assembly containing the type. When deserializing an object, formatters use this information to know what type of object to construct and initialize. ISerializationSurrogate allows you to define a surrogate type that takes over the serialization/deserialization duties of an object. However, a type that implements ISerializationSurrogate is tied to a specific type in a specific assembly.
However, there can be scenarios where it is required than an object be deserialized into a different type than the one it was serialized into. For example, you create a new version of a type and you want (for backwards compatibility) to be able to deserialize any already-serialized objects into the new version of the type.
Class System. Runtime. Serialization. SerializationBinder makes deserializing an object into a different type very easy. The general approach is this: define a type that derives from SerializationBinder and then set the formatter's Binder property to this SerializationBinde r-derived object. After setting the Binder property, call the formatter's Deserialize method.
During deserialization, the formatter determines that a binder has been set. As each object is about to be deserialized, the formatter calls the binder's BindToType method, passing it the assembly name and the type that the formatter want to deserialize. At this point, BindToType decides what type should actually be constructed and returns this type. Note that if the new type uses simple serialization via the [Serializable] attribute, then the original type and the new type must have the same exact field names and types. This restriction disappears if the new type implements ISerializable interface. This approach is shown below:
public class TypeXToTypeZBinder : System. Runtime. Serialization. SerializationBinder.
public override Type BindToType(string assemblyName, string typeName)
string strAssemblyXName = "AssemblyX, Version=1.0.0.0, Culture=neutral, PublicKeyToken=1234567890abcdef";
string strAssemblyXType = "ClassX";
string strAssemblyYType = "ClassY";
Type obTypeToDeserialize = null;
if ((assemblyName == strAssemblyXName) && (typeName == strAssemblyXType))
obTypeToDeserialize = Type. GetType( strAssemblyYName );
public void TestBinder()
System. Runtime. Serialization. IFormatter formatter = new BinaryFormatter();
formatter. Binder = new TypeXToTypeZBinder();
// to an object of type ClassY.
ClassY ob = (ClassY)formatter. Deserialize( stream );
Delegates and Serialization.
All delegates are compiles into serializable classes. This means that when serializing an object that contains a delegate member variable, the delegate's internal invocation list is serialized too. This can make serializing delegates very tricky as there are no guarantees that the target objects in the delegate's internal invocation list are also serializable. Consequently, serializing objects that contain delegates will work and sometimes it will through an exception. In addition, the object containing the delegate does not know/care about the state of the delegate. For example, this is the case when the delegate is used to manage event subscriptions - the exact number and identity of subscribers are often transient values that should persist during application sessions. Therefore, always mark delegate member variables as non-serializable using the [NonSerialized] attribute :
public class MyClass.
public delegate void delAlaram( DateTime dt );
// underlying delegate rather than to the event itself.
public event delAlaram evtAlarm;
Binary Formatter and Serialization Events.
Framework 2.0 introduces support for serialization events. Designated methods on your class will be called when serialization and deserialization takes place. Framework 2.0 defines four serialization and deserialization events - Serializing , Serialized , Deserializing , Deserialized . You designate methods as serialization/deserialization event handlers using method attributes as shown below:

Cadeia de Opções.
Not all stocks have options listed for trading. There are some criterias that the public company will need to meet before their stock options can be listed for trading. To find out whether options are available for trading, the simplest way is to enter the stock ticker symbol to retrieve the stock quote information and find out if there is a corresponding options chain available. The availability of an options chain will mean that there are options being traded for that stock.
The options chain shows the available call and put strike prices for a specific underlying security and expiration month. Depending on the online brokerage service that you use, the interface may be slightly different but in general, the layout and available information should be very similar.
Below is the options chain interface from OptionsXpress. The most important information is shown right at the top and they are usually the underlying security, along with its latest market price, and the expiration month. This is common sense as you don't want to purchase an option only to realise that its for the wrong underlier or the wrong expiration month!
Chamadas e Coloca.
Calls are usually listed on the left hand side while puts are typically displayed on the right hand side. In-the-money options are usually highlighted to differentiate them from out-of-the-money options. If you wish to trade at-the-money or near-the-money options, they are positioned on either side of the horizontal 'border' created by the highlighting.
O preço de greve.
Down the middle is the range of strike prices available for trading for the selected expiration month. The strike price intervals vary and depends on the price of the underlying. For lower priced stocks (usually $25 or less), intervals are at 2.5 points. Higher priced stocks have strike price intervals of 5 point (or 10 points for very expensive stocks priced at $200 or more).
Options Symbol.
Option symbols are unique to every option contract and they denote the type of option, the underlying asset and the expiration month, provided you have a good understanding of options symbology. However, they are seldom used nowadays since with modern computer technology, these information are often presented to the trader in a user friendly interface - the options chain! While you can enter the symbol directly when placing an order, it is advisable to select the desired options using the options chain interface to minimize human error.
Last Done Price.
The last done price reflects the latest transacted price for the specific option. As the most recent transaction may be hours or days ago, especially for thinly traded contracts, you should check the bid-ask price rather than the last done price to get a better picture of the current market value of the option you wish to trade.
Propagação Bid-Ask.
The bid and ask shows the price at which buyers are willing to pay and sellers are looking to receive for the particular option. The bid-ask spread is the difference between the bid and the ask and the size of the spread depends on the liquidity of the option. As a general rule, the lower the open interest, the wider the bid-ask spread. Furthermore, near the money options usually have higher open interest and hence better liquidity and narrower bid-ask spreads.
Você pode gostar.
Continue lendo.
Comprar Straddles em ganhos.
Comprar straddles é uma ótima maneira de jogar ganhos. Muitas vezes, a diferença no preço das ações subiu ou desceu após o relatório trimestral de ganhos, mas muitas vezes a direção do movimento pode ser imprevisível. Por exemplo, uma venda pode ocorrer mesmo que o relatório de ganhos seja bom se os investidores esperassem excelentes resultados. [Leia. ]
A escrita permite comprar estoques.
Se você é muito otimista em um estoque específico para o longo prazo e está procurando comprar o estoque, mas sente que está um pouco sobrevalorizado no momento, então você pode querer considerar escrever opções no estoque como um meio para adquiri-lo em um desconto. [Leia. ]
O que são opções binárias e como negociá-las?
Também conhecidas como opções digitais, as opções binárias pertencem a uma classe especial de opções exóticas em que o operador da opção especula puramente na direção do subjacente em um período de tempo relativamente curto. [Leia. ]
Investir em estoques de crescimento usando opções LEAPS®.
Se você está investindo no estilo de Peter Lynch, tentando prever o próximo multi-bagger, então você gostaria de saber mais sobre LEAPS® e por que considero que eles são uma ótima opção para investir no próximo Microsoft®. [Leia. ]
Efeito dos dividendos no preço das opções.
Os dividendos em dinheiro emitidos por ações têm grande impacto nos preços das opções. Isso ocorre porque o preço do estoque subjacente deve cair pelo valor do dividendo na data do ex-dividendo. [Leia. ]
Bull Call Spread: uma alternativa para a chamada coberta.
Como alternativa à escrita de chamadas cobertas, pode-se inserir um spread de call bull para um potencial de lucro semelhante, mas com um requisito de capital significativamente menor. Em vez de manter o estoque subjacente na estratégia de chamadas cobertas, a alternativa. [Leia. ]
Captura de dividendos usando chamadas cobertas.
Algumas ações pagam dividendos generosos a cada trimestre. Você qualifica o dividendo se você estiver segurando as ações antes da data do ex-dividendo. [Leia. ]
Aproveite as chamadas, não Margin Calls.
Para obter retornos mais altos no mercado de ações, além de fazer mais lição de casa nas empresas que deseja comprar, muitas vezes é necessário assumir maior risco. Uma maneira mais comum de fazer isso é comprar ações na margem. [Leia. ]
Day Trading usando Opções.
As opções de negociação do dia podem ser uma estratégia bem sucedida e rentável, mas há algumas coisas que você precisa saber antes de usar começar a usar opções para o dia comercial. [Leia. ]
Qual é a relação de chamada de chamada e como usá-la.
Saiba mais sobre a proporção de apontar, a forma como ela é derivada e como ela pode ser usada como um indicador contrário. [Leia. ]
Compreender a paridade de colocação de chamadas.
A paridade de chamada de compra é um princípio importante no preço de opções identificado pela primeira vez por Hans Stoll em seu artigo, The Relation Between Put and Call Prices, em 1969. Ele afirma que o prémio de uma opção de compra implica um certo preço justo para a opção de venda correspondente com o mesmo preço de exercício e data de vencimento, e vice-versa. [Leia. ]
Compreendo os gregos.
Na negociação de opções, você pode notar o uso de certos alfabetos gregos como delta ou gama ao descrever os riscos associados a várias posições. Eles são conhecidos como "os gregos". [Leia. ]
Valorizando ações comuns usando a análise de fluxo de caixa descontada.
Uma vez que o valor das opções de compra de ações depende do preço do estoque subjacente, é útil calcular o valor justo das ações usando uma técnica conhecida como fluxo de caixa descontado. [Leia. ]
Siga-nos no Facebook para obter estratégias diárias e amp; Dicas!
Opções básicas.
Estratégias de opções.
Opções Estratégia Finder.
Aviso de Risco: as ações, futuros e negociação de opções binárias discutidas neste site podem ser consideradas Operações de Negociação de Alto Risco e sua execução pode ser muito arriscada e pode resultar em perdas significativas ou mesmo em uma perda total de todos os fundos em sua conta. Você não deve arriscar mais do que você pode perder. Antes de decidir comercializar, você precisa garantir que compreenda os riscos envolvidos levando em consideração seus objetivos de investimento e nível de experiência. As informações contidas neste site são fornecidas apenas para fins informativos e educacionais e não se destinam a ser um serviço de recomendação comercial. TheOptionsGuide não será responsável por erros, omissões ou atrasos no conteúdo, ou por quaisquer ações tomadas com base nisso.
Os produtos financeiros oferecidos pela empresa possuem alto nível de risco e podem resultar na perda de todos os seus fundos. Você nunca deve investir dinheiro que não pode perder.

CodeKata.
Because experience.
How do you get to be a great musician? It helps to know the theory, and to understand the mechanics of your instrument. It helps to have talent. But ultimately, greatness comes from practicing; applying the theory over and over again, using feedback to get better every time.
How do you get to be an All-Star sports person? Obviously fitness and talent help. But the great athletes spend hours and hours every day, practicing.
But in the software industry we take developers trained in the theory and throw them straight in to the deep-end, working on a project. It’s like taking a group of fit kids and telling them that they have four quarters to beat the Redskins (hey, we manage by objectives, right?). In software we do our practicing on the job, and that’s why we make mistakes on the job. We need to find ways of splitting the practice from the profession. We need practice sessions.
What makes a good practice session? You need time without interruptions, and a simple thing you want to try. You need to try it as many times as it takes, and be comfortable making mistakes. You need to look for feedback each time so you can work to improve. There needs to be no pressure: this is why it is hard to practice in a project environment. it helps to keep it fun: make small steps forward when you can. Finally, you’ll recognize a good practice session because you’ll came out of it knowing more than when you went in.
Code Kata is an attempt to bring this element of practice to software development. A kata is an exercise in karate where you repeat a form many, many times, making little improvements in each. The intent behind code kata is similar. Each is a short exercise (perhaps 30 minutes to an hour long). Some involve programming, and can be coded in many different ways. Some are open ended, and involve thinking about the issues behind programming. These are unlikely to have a single correct answer. I add a new kata every week or so. Invest some time in your craft and try them.
Remember that the point of the kata is not arriving at a correct answer. The point is the stuff you learn along the way. The goal is the practice, not the solution.
The Kata.
Kata 1: Supermarket pricing. Pricing looks easy, but scratch the surface and there are some interesting issues to consider.
Kata 2: Karate Chop. A binary chop algorithm is fairly boring. Until you have to implement it using five totally different techniques.
Kata 3: How Big, How Fast? Quick estimation is invaluable when it comes to making design and implementation decisions. Here are some questions to make you turn over the envelope.
Kata 4: Data Munging. Implement two simple data extraction routines, and see how much they have in common.
Kata 5: Bloom Filters. Implement a simple hash-based lookup mechanism and explore its characteristics.
Kata 6: Anagrams. Find all the anagram combinations in a dictionary.
Kata 7: Reviewing. What does our code look like through critical eyes, and how can we make our eyes more critical?
Kata 8: Objectives. What effects do our objectives have on the way we write code?
Kata 9: Checkout. Back to the supermarket. This week, we’ll implement the code for a checkout system that handles pricing schemes such as “apples cost 50 cents, three apples cost $1.30.”
Kata 10: Hash vs. Class. Is it always correct to use (for example) classes and objects to structure complex business objects, or couple simpler structures (hash as Hashes) do the job?
Kata 11: Sorting it Out. Just because we need to sort something doesn’t necessarily mean we need to use a conventional sorting algorithm.
Kata 12: Best Sellers. Consider the implementation of a top-ten best sellers list for a high volume web store.
Kata 13: Counting Lines. Counting lines of code in Java source is not quite as simple as it seems.
Kata 14: Trigrams. Generating text using trigram analysis lets us experiment with different heuristics. (And it let’s us create new, original Tom Swift stories…)
Kata 15: Playing with bits. A diversion to discover the pattern in some bit sequences.
Kata 16: Business Rules. How can you tame a wild (and changing) set of business rules?
Kata 17: More Business Rules. The rules that specify the overall processing of an order can be complex too, particularly as they often involve waiting around for things to happen.
Kata 18: Dependencies. Let’s write some code that calculates how dependencies propagate between things such as classes in a program.
Kata 19: Word chains. Write a program that solves word chain puzzles (cat в†’ cot в†’ dot в†’ dog).
Kata 20: Klondike. Experiment with various heuristics for playing the game Klondike.
Kata 21: Simple Lists. Play with different implementations of a simple list.
I have to admit that I’m nervous doing this. My hope is that folks will work on the kata for a while before discussing them; much of the benefit comes from the little “a-ha!” moments along the way. So, it’ll be interesting to see how (and if) the discussion develops.
CodeKata: How It Started.
(This is a long one. It explains how I discovered that something I do almost every day to improve my coding is actually a little ritual that has much in common with practice in the martial arts…)
Kata, Kumite, Koan, and Dreyfus.
A week or so ago I posted a piece called CodeKata, suggesting that as developers we need to spend more time just practicing: writing throwaway code just to get the experience of writing it. I followed this up with a first exercise, an experiment in supermarket pricing.
Kata01: Supermarket Pricing.
This kata arose from some discussions we’ve been having at the DFW Practioners meetings. The problem domain is something seemingly simple: pricing goods at supermarkets.
Kata02: Karate Chop.
A binary chop (sometimes called the more prosaic binary search) finds the position of value in a sorted array of values. It achieves some efficiency by halving the number of items under consideration each time it probes the values: in the first pass it determines whether the required value is in the top or the bottom half of the list of values. In the second pass in considers only this half, again dividing it in to two. It stops when it finds the value it is looking for, or when it runs out of array to search. Binary searches are a favorite of CS lecturers.
Kata03: How Big? How Fast?
Rough estimation is a useful talent to possess. As you’re coding away, you may suddenly need to work out approximately how big a data structure will be, or how fast some loop will run. The faster you can do this, the less the coding flow will be disturbed.
Kata04: Data Munging.
Martin Fowler gave me a hard time for Kata02, complaining that it was yet another single-function, academic exercise. Which, or course, it was. So this week let’s mix things up a bit.
Here’s an exercise in three parts to do with real world data. Try hard not to read ahead—do each part in turn.
Kata05: Bloom Filters.
There are many circumstances where we need to find out if something is a member of a set, and many algorithms for doing it. If the set is small, you can use bitmaps. When they get larger, hashes are a useful technique. But when the sets get big, we start bumping in to limitations. Holding 250,000 words in memory for a spell checker might be too big an overhead if your target environment is a PDA or cell phone. Keeping a list of web-pages visited might be extravagant when you get up to tens of millions of pages. Fortunately, there’s a technique that can help.
Kata06: Anagrams.
Back to non-realistic coding this week (sorry, Martin). Let’s solve some crossword puzzle clues.
Kata07: How’d I Do?
The last couple of kata have been programming challenges; let’s move back into mushier, people-oriented stuff this week.
Postagens recentes.
Direitos autorais e cópia; 2016 - Dave Thomas (@PragDave) - Powered by Octopress.

Binary option chains

If you would like to learn CFTP and/or Fill's algorithm, there are several expositions to choose from. Many articles which develop new coupling techniques for use in CFTP or Fill's algorithm also contain brief explanations of these techniques. Propp-Wilson '96 describe CFTP. Fill '98 describes his new algorithm, and also gives an explanation of CFTP that some probabilists prefer. Sections 1 and 7 are good places to start reading. Propp '97 gives a survey of CFTP with a focus on combinatorial applications. Propp-Wilson '98 give a user's guide to CFTP. Dimakos '99 gives a guide to CFTP and Fill's algorithm. Thönnes '99 gives a primer on perfect simulation focussed on CFTP. Fill-Machida-Murdoch-Rosenthal '00 give a readable extension of Fill's algorithm. Wilson '00 section 1 gives a primer on CFTP and extensions. Häggström '00 wrote course notes (now a book) on ``Finite Markov chains and algorithmic applications'', including a chapter on CFTP. Casella-Lavine-Robert '00 give an explanation of CFTP and Fill's algorithm. Møller '00 gives an exposition of CFTP and Fill's algorithm with a focus on stochastic geometry. David J. C. MacKay '03 wrote a book (``Information Theory, Inference, and Learning Algorithms'') which contains a 9-page introduction to perfect sampling. There is a chapter on ``Coupling from the past,'' by James G. Propp and David B. Wilson, Chapter 22 of the textbook ''Markov Chains and Mixing Times,'' by David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, to be published by the American Mathematical Society, 2008.
Introduction and Scope.
Random sampling has found numerous applications in physics, statistics, and computer science. Perhaps the most versatile method of generating random samples from a probability space is to run a Markov chain. But for how many steps?
In most cases one simply does not know how many Markov chain steps are needed to get a sufficiently random state. There is a large literature of heuristic algorithms for inferring when enough steps have been taken, but they are non-rigorous, and one never knows for sure that an adequate number of steps have been taken. These heuristic algorithms are beyond the scope of this bibliography, but the interested reader is referred to some lecture notes by Sokal and the MCMC Preprint Service.
In the past decade there is been much research on obtaining rigorous bounds of how many Markov chain steps are needed to generate a random sample. Sometimes these bounds are tight, sometimes they are unduly pessimistic. The interested reader is referred to a survey by Diaconis and Saloff-Coste and a survey by Jerrum and Sinclair; the size of this literature makes it beyond the scope of this bibliography.
In recent years there have been a large number of algorithms developed for sampling from the steady state distribution of suitably well-structured Markov chains, which require no a priori knowledge of how long the Markov chains take to get mixed. The algorithms determine on their own, during run time, how many steps to run the Markov chain. It is these algorithms that are the focus of this bibliography. Most of these algorithms return samples that are distributed exactly according to the stationary distribution of the Markov chain, but a few return samples that have some bias ε that the user can make as small as desired. Since the focus of this bibliography is on working computer algorithms, the symbol is placed next to those articles that contain simulation results or give sample outputs. Each annotated entry contains links relevant to the paper, giving the article's abstract (click on the title) and authors' homepages when available, as well as links to online preprints. Several of the annotations were contributed by people other than the maintainer.
It has been pointed out that ``stationary stopping times'' may also be considered to be algorithms for sampling from a Markov chain's stationary distribution. However, these Markov chains typically have state spaces such as the symmetric group or the hypercube, for which one already knows how to effectively generate a random sample on a computer. Rather, the point of studying these stopping times is to understand interesting mathematical processes, such as shuffling a deck of cards. (A notable exception is Fill's algorithm.) Since the literature on these stopping times is sizable, only those articles with an algorithmic theme are included. The reader interested in stopping times is referred to the articles by Aldous and Diaconis and Diaconis and Fill, which contain many references.
Also relevant to the present bibliography is a literature on backwards compositions of random maps. The backwards composition of random maps can be used to define a ``stochastically recursive sequence'' of points from the state space. The articles from this literature study when this sequence of points converge almost surely, since convergence implies the existence of a stationary distribution for the Markov chain. (Existence can be nontrivial for infinite state spaces.) If one has the additional conditions that 1) the convergence occurs after a finite number steps, and 2) one can determine when this convergence has occurred, then one may use the technique of ``coupling from the past'', which is used in several the articles listed below, to generate random samples from the state space. (There are numerous examples in which conditions 1 or 2 do not hold.) Diaconis and Freedman give a survey of the literature on stochastically recursive sequences; a few representative articles are listed below.
Annotated Bibliographic Entries.
Andrei Broder. Generating random spanning trees. In 30th Annual Symposium on Foundations of Computer Science , pages 442--447, 1989.
David J. Aldous. A random walk construction of uniform spanning trees and uniform labelled trees. SIAM Journal on Discrete Mathematics , 3(4):450--465, 1990.
These two articles give the same independently discovered random-walk based algorithm for generating random spanning trees of a graph. The algorithm uses a Markov chain on the set of spanning trees of an undirected graph to return a (perfectly) random spanning tree. Broder uses the algorithm to analyze the random walk on a ring. Aldous uses the algorithm to determine the properties of random trees and to compute some non-trivial probabilities pertaining to the random walk in the plane. (There is another random tree algorithm based on computing determinants.) Søren Asmussen, Peter W. Glynn, and Hermann Thorisson. Stationary detection in the initial transient problem. ACM Transactions on Modeling and Computer Simulation , 2(2):130--157, 1992.
This paper explores what is possible and what is not, and was the first paper to show that it is possible to obtain unbiased samples from the steady state distribution of a finite Markov chain by observing it, provided it is irreducible and one knows how many states it has. Equivalently, there is a universal randomized stationary stopping time that works for all (irreducible) Markov chains with a given (finite) number of states. David J. Aldous. On simulating a Markov chain stationary distribution when transition probabilities are unknown. In D. J. Aldous, P. Diaconis, J. Spencer, and J. M. Steele, editors, Discrete Probability and Algorithms , volume 72 of IMA Volumes in Mathematics and its Applications , pages 1--9. Springer-Verlag, 1995.
Abstract: We present an algorithm which, given a n - state Markov chain whose steps can be simulated, outputs a random state whose distribution is within ε of the stationary distribution, using O(n) space and O (ε -2 τ) time, where τ is a certain ``average hitting time" parameter of the chain.
Remarks: While not exact, this algorithm was much more efficient than the previous one, and directly stimulated the development of two subsequent exact sampling algorithms. This paper gives the only nontrivial lower bound on the running time of an algorithm for exact sampling from generic Markov chains. Dana Randall and Alistair Sinclair. Testable algorithms for self-avoiding walks. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms , pages 593-602, 1994. (postscript)
Abstract: We present a polynomial time Monte Carlo algorithm for almost uniformly generating and approximately counting self-avoiding walks in rectangular lattices Z d . These are classical problems that arise, for example, in the study of long polymer chains. While there are a number of Monte Carlo algorithms used to solve these problems in practice, these are heuristic and their correctness relies on unproven conjectures. In contrast, our algorithm depends on a single, widely-believed conjecture that is weaker than preceding assumptions, and, more importantly, is one which the algorithm itself can test. Thus our algorithm is reliable , in the sense that it either outputs answers that are guaranteed, with high probability, to be correct, or finds a counter-example to the conjecture. László Lovász and Peter Winkler. Exact mixing in an unknown Markov chain. Electronic Journal of Combinatorics , 2, 1995. Paper #R15.
Abstract: We give a simple stopping rule which will stop an unknown, irreducible $n$-state Markov chain at a state whose probability distribution is exactly the stationary distribution of the chain. The expected stopping time of the rule is bounded by a polynomial in the maximum mean hitting time of the chain. Our stopping rule can be made deterministic unless the chain itself has no random transitions.
Remarks: This paper gives the first (universal) exact sampling algorithm that runs in time that is polynomial in certain parameters associated with the Markov chain. Gives a deterministic stationary stopping time that works when the Markov chain itself is not deterministic. The paper also contains a pretty lemma on random trees that is of independent interest. James G. Propp and David B. Wilson. Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms , 9(1&2):223--252, 1996.
Abstract: For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i. e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the algorithm itself. If the state space has a partial order that is preserved under the moves of the Markov chain, then the coupling is often particularly efficient. Using our approach one can sample from the Gibbs distributions associated with various statistical mechanics models (including Ising, random-cluster, ice, and dimer) or choose uniformly at random from the elements of a finite distributive lattice.
Remarks: This paper gives an algorithm, monotone coupling from the past, for exact sampling with Markov chains on huge state spaces. Since monotone-CFTP relies on the Markov chain having special structure, it is not ``universal'' as several of the above algorithms are, but it is practical for a surprising number of previously studied Markov chains. Includes simulation results for the random cluster and dimer models. Valen E. Johnson. Studying convergence of Markov chain Monte Carlo algorithms using coupled sample paths. Journal of the American Statistical Association , 91(433):154--166, 1996. (Postscript.)
Abstract: I describe a simple procedure for investigating the convergence properties of Markov Chain Monte Carlo sampling schemes. The procedure employs multiple runs from a sampler, using the same random deviates for each run. When the sample paths from all sequences converge, it is argued that approximate equilibrium conditions hold. The procedure also provides a simple diagnostic for detecting modes in multimodal posteriors. Several examples of the procedure are provided. In Ising models, the relation between the correlation parameter and the convergence rate of rudimentary Gibbs samplers is investigated. In another example, the effects of multiple modes on the convergence of coupled paths are explored using mixtures of bivariate normal distributions. The technique is also used to evaluate the convergence properties of a Gibbs sampling scheme applied to a model for rat growth rates (Gelfand et al 1990).
Remarks: While technically not a paper on exact sampling, this paper investigates how the mixing time of a Markov chain may be inferred by running a large number of coupled simulations until they coalesce. The initial states of the Markov chains are chosen at random, and if the probability of rejection in rejection sampling is known, then rigorous estimates of the mixing time are given. Includes the (independently made) observation that for monotone Markov chains, only two coupled states need to be simulated. Includes simulation results. Additional articles that take a similar approach are available from Johnson's homepage. Michael Luby, Dana Randall, and Alistair Sinclair. Markov chain algorithms for planar lattice structures (extended abstract). In 36th Annual Symposium on Foundations of Computer Science , pages 150--159, 1995.
Abstract: Consider the following Markov chain, whose states are all domino tilings of a 2 n X 2 n chessboard: starting from some arbitrary tiling, pick a 2 X 2 window uniformly at random. If the four squares appearing in this window are covered by two parallel dominoes, rotate the dominoes in place. Repeat many times. This process is used in practice to generate a random tiling, and is a key tool in the study of the combinatorics of tilings and the behavior of dimer systems in statistical physics. Analogous Markov chains are used to randomly generate other structures on various two-dimensional lattices. This paper presents techniques which prove for the first time that, in many interesting cases, a small number of random moves suffice to obtain a uniform distribution.
Remarks: This paper gives three new Markov chains for sampling certain dimer and ice systems. The focus of this paper is provable running time bounds. Monotone-CFTP may be applied to each of their Markov chains to get an exact algorithm; when this is done, their proofs may be interpreted as a priori bounds on the running time of CFTP, though in practice the exact algorithm runs much more quickly than the bounds suggest.
Remarks: For the particular case of the 2 n X 2 n chessboard, it is much faster to generate a random spanning tree, and then use a Temperley-like bijection to convert it to a (perfectly) random domino tiling. Their methods apply to more general regions for which such a Temperleyan bijection does not exist. James G. Propp and David B. Wilson. How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph. Journal of Algorithms , 27:170--217, 1998. This article combines two conference papers, appearing in Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms and Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing .
Abstract: A general problem in computational probability theory is that of generating a random sample from the state space of a Markov chain in accordance with the steady-state probability law of the chain. Another problem is that of generating a random spanning tree of a graph or spanning arborescence of a directed graph in accordance with the uniform distribution, or more generally in accordance with a distribution given by weights on the edges of the graph or digraph. This paper gives algorithms for both of these problems, improving on earlier results and exploiting the duality between the two problems. Each of the new algorithms hinges on the recently-introduced technique of coupling from the past or on the linked notions of loop-erased random walk and ``cycle-popping''.
Remarks: This article gives what are so far the best algorithms for exact sampling on generic Markov chains, as well as the first application of CFTP to a huge non-monotone state space. It gives a universal randomized stationary stopping time that is within a constant factor of optimal, and another algorithm that is faster when the Markov chain can be simulated starting from any state rather than just observed in action. Surprisingly, both algorithms are intimately related to the generation of random spanning trees of a weighted directed graph, and this paper gives tree algorithms that are both faster and more general than the Broder/Aldous algorithm. The algorithms use various versions of CFTP (voter-CFTP, coalescing-CFTP, cover-CFTP, and tree-CFTP), and another technique called cycle popping. (An upcoming book by Lyons and Peres makes use of the cycle-popping algorithm when analyzing essential spanning forests of infinite graphs.) Includes a sample output from the tree algorithm. David Eppstein. Representing all minimum spanning trees with applications to counting and generation, Technical Report 95-50, U. C. Irvine, 1995.
Shows how to use the algorithms for random spanning tree generation to solve other random sampling problems. Per Eppstein's description: Shows how to find for any edge weighted graph G an equivalent graph EG such that the minimum spanning trees of G correspond one-for-one with the spanning trees of EG. The equivalent graph can be constructed in time O(m+n log n) given a single minimum spanning tree of G. As a consequence one can find fast algorithms for counting, listing, and randomly generating MSTs. Also discusses similar equivalent graph constructions for shortest paths, minimum cost flows, and bipartite matching. Stefan Felsner and Lorenz Wernisch. Markov chains for linear extensions, the two-dimensional case. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms , pages 239--247, 1997. (full version)
Abstract: We study the generation of uniformly distributed linear extensions using Markov chains. In particular we show that monotone coupling from the past can be applied in the case of linear extensions of two-dimensional orders. For width two orders a mixing rate of O(n^3 log n) is proved. We conjecture that this is the mixing rate in the general case and support the conjecture by empirical data. On the course we obtain several nice structural results concerning Bruhat order and weak Bruhat order of permutations.
Remarks: Subsequent work confirmed their conjecture. Wilfrid S. Kendall. Perfect simulation for the area-interaction point process. In L. Accardi and C. C. Heyde, editors, Probability Towards 2000 , pages 218--234. Springer, 1998.
Abstract: The ideas of Propp and Wilson ("Exact sampling with coupled Markov chains and applications to statistical mechanics", Random Structures and Algorithms, 1996, 9:223-252) for exact simulation of Markov chains are extended to deal with perfect simulation of attractive area-interaction point processes in bounded windows. A simple modification of the basic algorithm is described which provides perfect simulation of the non-monotonic case of the repulsive area-interaction point process. Results from simulations using a C computer program are reported; these confirm the practicality of this approach in both attractive and repulsive cases. The paper concludes by mentioning other point processes which can be simulated perfectly in this way, and by speculating on useful future directions of research.
Remarks: Shows how to do perfect simulations of the area-interaction point process, though the techniques extend to more general point processes. A unique feature of this application is that the Markov chain used is not uniformly ergodic, i. e. its mixing time is infinite. (The state space is an infinite partially ordered set with no top state, and one can find states from which the Markov chain takes arbirtrarily long to equilibrate.) To deal with this problem, Kendall modified monotone-CFTP, replacing references to the (nonexistent) top state with references to a stochastically dominating process. Kendall also shows how to apply CFTP to repulsive point processes, which are anti-monotone. Includes simulation results. Wilfrid S. Kendall. On some weighted Boolean models. In D. Jeulin, editor, Advances in Theory and Applications of Random Sets , pages 105--120. World Scientific Publishing Company, 1997. (postscript)
Following up on the previous article, shows how to apply CFTP to attractive birth-death processes and exclusion processes. James A. Fill. An interruptible algorithm for perfect sampling via Markov chains. The Annals of Applied Probability , 8(1):131--162, 1998. (Postscript.) An extended abstract appeared in the Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing .
Abstract: For a large class of examples arising in statistical physics known as attractive spin systems (e. g., the Ising model), one seeks to sample from a probability distribution π on an enormously large state space, but elementary sampling is ruled out by the infeasibility of calculating an appropriate normalizing constant. The same difficulty arises in computer science problems where one seeks to sample randomly from a large finite distributive lattice whose precise size cannot be ascertained in any reasonable amount of time. The Markov chain Monte Carlo (MCMC) approximate sampling approach to such a problem is to construct and run ``for a long time'' a Markov chain with long-run distribution π. But determining how long is long enough to get a good approximation can be both analytically and empirically difficult. Recently, Jim Propp and David Wilson have devised an ingenious and efficient algorithm to use the same Markov chains to produce perfect (i. e., exact) samples from π. However, the running time of their algorithm is an unbounded random variable whose order of magnitude is typically unknown a priori and which is not independent of the state sampled, so a naive user with limited patience who aborts a long run of the algorithm will introduce bias. We present a new algorithm which (1) again uses the same Markov chains to produce perfect samples from π, but is based on a different idea (namely, acceptance/rejection sampling); and (2) eliminates user-impatience bias. Like the Propp-Wilson algorithm, the new algorithm applies to a general class of suitably monotone chains, and also (with modification) to ``anti-monotone'' chains. When the chain is reversible, naive implementation of the algorithm uses fewer transitions but more space than Propp-Wilson. When fine-tuned and applied with the aid of a typical pseudorandom number generator to an attractive spin system on n sites using a random site updating Gibbs sampler whose mixing time tau is polynomial in n, the algorithm runs in time of the same order (bound) as Propp-Wilson [expectation O(tau log n)] and uses only logarithmically more space [expectation O(n log n), vs. O(n) for Propp-Wilson]. Peter W. Glynn and Philip Heidelberger. Bias properties of budget constrained simulations. Operations Research , 38(5):801--814, 1990.
Among other things, this article shows that so long as the user insists on waiting long enough to get at least one random sample, a deadline will not introduce bias. This holds for any sampling procedure, whether the deadline is in terms of Markov chain steps, real time, or any other measure. James A. Fill. The move-to-front rule: a case study for two exact sampling algorithms. Probability in the Engineering and Informational Sciences , 12:283--302, 1998.
Abstract: The elementary problem of exhaustively sampling a finite population without replacement is used as a nonreversible test case for comparing two recently proposed MCMC algorithms for perfect sampling, one based on backward coupling and the other on strong stationary duality. The backward coupling algorithm runs faster in this case, but the duality-based algorithm is unbiased for user impatience. An interesting by-product of the analysis is a new and simple stochastic interpretation of a mixing-time result for the move-to-front rule.
Remark: See also this paper. O. Häggström, M. N. M. van Lieshout, and J. Møller. Characterisation results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes. Bernoulli , 5(4):641--658, 1999.
Abstract: The area-interaction process and the continuum random-cluster model are characterised in terms of certain functional forms of their respective conditional intensities. In certain cases, these two point process models can be derived from a bivariate point process model which in many respects is simpler to analyse and simulate. Using this correspondence we devise a two-component Gibbs sampler, which can be used for fast and exact simulation by extending the recent ideas of Propp and Wilson. We further introduce a Swendsen-Wang type algorithm. The relevance of the results within spatial statistics as well as statistical physics is discussed.
Remarks: Shows how to apply monotone-CFTP to sample from the Widom-Rowlinson point process. Here there is no top or bottom state, but (in contrast to Kendall's chain) such states can be adjoined to the state space. (The authors instead use two other states, already in the state space, that are just as effective at testing coalescence.) The Widom-Rowlinson model can be marginalized to obtain the attractive area-interaction point process and the continuum random cluster model (with positive integral q ). Sampling from the area-interaction process in this way turns out to be faster than (though not as generalizable as) Kendall's approach. The paper also describes a Swendsen-Wang type algorithm, and includes simulation results. Jesper Møller. Markov chain Monte Carlo and spatial point processes. In W. S. Kendall, O. E. Barndorff-Nielsen, and M. N. M. van Lieshout, editors, Stochastic Geometry: Likelihood and Computation , Monographs on Statistics and Applied Probability #80, pages 141--172. Chapman and Hall / CRC Press, 1998.
Abstract: This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using ``coupling from the past'' to remove initialization bias from Monte Carlo randomization. The article describes several applications to specific kinds of combinatorial objects such as tilings, constrained lattice paths, an alternating sign matrices. Olle Häggström and Karin Nelander. Exact sampling from anti-monotone systems. Statistica Neerlandica , 52:360--380, 1998. Postscript.
Abstract: A new approach to Markov chain Monte Carlo simulation was recently proposed by Propp and Wilson. This approach, unlike traditional ones, yield samples which have exactly the desired distribution. The Propp-Wilson algorithm requires this distribution to have a certain structure called monotonicity. In this paper an idea of Kendall is applied to show how the algorithm can be extended to the case where monotonicity is replaced by anti-monotonicity. As illustrating examples, simulations of the hard-core model and the random-cluster model are presented. Michael Luby and Eric Vigoda. Approximately counting up to four (extended abstract). In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing , pages 682--687, 1997. (Postscript.)
Abstract: We present a fully-polynomial scheme to approximate the number of independent sets in graphs with maximum degree four. In general, for graphs with maximum degree Δ≥4, the scheme approximates a weighted sum of independent sets. The weight of each independent set is expressed in terms of a positive parameter λ≤ 1/(Δ-3), where the weight of independent set S is λ |S| . We also prove complementary hardness of approximation results, which show that it is hard to approximate the weighted sum for values of λ & gt; c /Δ for some constant c > 0.
Remarks: Gives a new Markov chain for generating random independent sets with activity λ. When the maximum degree Δ≥4 and λ ≤ 1/(Δ-3) , they show that this Markov chain is rapidly mixing. CFTP may be efficiently applied to their Markov chain; when this is done, their proofs (but not the theorem itself) may be interpreted as a priori bounds on the running time of CFTP.
Abstract: Propp and Wilson (1995) described a protocol, called coupling from the past, for exact sampling from a finite distribution using a coupled Markov chain Monte Carlo algorithm. In this paper we extend coupling from the past to various MCMC samplers on a continuous state space; rather than following the monotone sampling device of Propp and Wilson, our approach uses methods related to gamma-coupling and rejection sampling to simulate the chain, and direct accounting of sample paths to check whether they have coupled.
Remarks: Describes a variety of scenarios for which CFTP may be applied to a (non-monotone) continuous state space, giving a sequence of algorithms that start out simple, but become increasingly sophisticated. The methods used are related to gamma-coupling and rejection sampling, and appear to be applicable to Bayesian parameter estimation. Includes a case study comparing the performance of each technique when used to sample from the posterior distribution of a set of pump reliability parameters. Wilfrid S. Kendall and Jesper Møller. Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Advances in Applied Probability 32(3):844--865, 2000. (Postscript)
Abstract: In this paper we investigate the application of perfect simulation, in particular Coupling from The Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatial birth-and-death processes. We then investigate discrete-time Metropolis-Hastings samplers for point processes, and show how a variant which samples systematically from cells can be converted into a perfect version. An application is given to the Strauss point process.
See also the document describing the implementation. Haiyan Cai. A note on an exact sampling algorithm and Metropolis Markov chains. Technical report, University of Missouri, St. Louis, 1997.
Shows how to take a generic probability distribution on a space, together with a reference distribution from which it is already known how to sample, and construct a certain particular Metropolis-type monotone Markov chain. Coalescence occurs precisely when the rejection sampler with the same reference measure would accept. S. G. Foss and R. L. Tweedie. Perfect simulation and backward coupling. Stochastic Models , 14(1-2):187--203, 1998. (Postscript.)
Abstract: Algorithms for perfect or exact simulation of random samples from the invariant measure of a Markov chain have received considerable recent attention following the introduction of the "coupling-from-the-past" (CFTP) technique of Propp and Wilson. Here, we place such algorithms in the context of backward coupling of stochastically recursive sequences. We show that although general backward couplings can be constructed for chains with finite mean forward coupling times, and can even be thought of as extending the classical "Loynes schemes" from queuing theory, successful "vertical" CFTP algorithms such as those of Propp and Wilson can be constructed if and only if the chain is uniformly geometric ergodic. We also relate the convergence moments for backward coupling methods to those of forward coupling times: the former typically lose at most one moment compared to the latter.
Remarks: Relates CFTP to more general stochastically recursive sequences, for which one has a sequence of random variables that with probability one converge to a particular value with the right distribution, but for which one isn't necessarily able to determine when this convergence has happened. Studies the moments of the convergence time of stochastically recursive sequences, some of which may be finite with others being infinite. Robert B. Lund and David B. Wilson. Exact sampling algorithms for storage systems. In preparation.
Show how to apply monotone-CFTP to sample from the steady-state distribution of certain storage systems. Rather than the linear data-flow used in traditional monotone-CFTP, the flow of data through the algorithm is two-dimensional. Jesper Møller. Perfect simulation of conditionally specified models. Journal of the Royal Statistical Society, Ser. B , 61(1):251--264, 1999.
Abstract: We discuss how the ideas of producing perfect simulations based on coupling from the past for finite state space models naturally extend to multivariate distributions with infinite or uncountable state spaces such as auto-gamma, auto-Poisson and autonegative binomial models, using Gibbs sampling in combination with sandwiching methods originally introduced for perfect simulation of point processes.
Remarks: Shows how to apply CFTP to sample from a class of Markov random fields, in which the individual variables (from a finite, countable, or continuous space) depend upon one another in a monotone or anti-monotone way. For the continuous distributions, the user specifies an ε, which controls not the bias (which is 0), but instead the numerical precision of the output. The application to the auto-gamma distribution, which includes the pump-reliability application above, appears to be faster than the approach of Murdoch and Green.
For the continuous distributions there is an even faster method, which also takes the numerical error ε to zero.
The aforementioned sandwiching methods are the ones introduced for antimonotone CFTP by Kendall and Häggström-Nelander. W. S. Kendall. Perfect Simulation for Spatial Point Processes. In Bulletin of the International Statistical Institute 51 st Session, Istanbul (August 1997) , volume 3, pages 163--166, 1997.
Abstract: A survey is given of the ideas of perfect simulation involving Coupling From The Past (CFTP) as introduced by Propp and Wilson (1996), and its application to the perfect simulation of spatial point processes. A sketch is given of an extension to the perfect simulation of point processes defined over all R^2.
Sommaire: Voici un exposй des idйes de la simulation parfaite au moyen de Coupling From The Past (CFTP) selon Propp et Wilson (1996), et son application а la simulation des processus ponctuels en espace. Une indication est fournie du moyen de conduire la simulation parfaite des processus ponctuels йtendus а travers tout R^2. S. G. Foss, R. L. Tweedie, and J. N. Corcoran. Simulating the invariant measures of Markov chains using backward coupling at regeneration times. Probability in the Engineering and Informational Sciences , 12:303--320, 1998. (postscript).
Abstract: We develop an algorithm for simulating approximate random samples from the invariant measure of a Markov chain using backward coupling of embedded regeneration times. Related methods have been used effectively for finite chains and for stochastically monotone chains: here we propose a method of implementation which avoids these restrictions by using a ``cycle-length'' truncation. We show that the coupling times have good theoretical properties and describe benefits and difficulties of implementing the methods in practice.
Remarks: Gives a method for approximately sampling from the steady-state distribution of a Markov chain when the state of the chain takes on a certain particular value on a semi-regular basis. The Markov chain of interest is lifted to one that keeps track of 1) the original chain's state, and 2) the number of steps since the original chain's state took on that special value. The lifted chain is then truncated to force a return to the special state whenever the second coordinate gets too big. Using an independent set of coins for each value of time and each value of the second coordinate of the modified lifted chain, the authors show how to effectively test for coalescence. Includes discussion for how to choose the truncation parameter, though this not mechanized to the extent where the user could specify a desired maximum bias. Includes a case study on a two-server re-entrant queueing network. Elke Thönnes. Perfect simulation of some point processes for the impatient user. Advances in Applied Probability, Stochastic Geometry and Statistical Applications , 31(1):69--87, 1999. (Postscript.)
Abstract: Recently Propp and Wilson have proposed an algorithm, called Coupling from the Past (CFTP), which allows not only an approximate but perfect (i. e. exact simulation of the stationary distribution of certain finite state space Markov chains. Perfect Sampling using CFTP has been successfully extended to the context of point processes, amongst other authors, by Häggström et al.. In Häggström et al. Gibbs Sampling is applied to a bivariate point process, the pentrable spheres mixture model. Fill also introduced an exact sampling algorithm, which, in contrast to CFTP, is unbiased for user impatience. Fill's algorithm is a form of rejection sampling and similar to CFTP requires sufficient monotonicity properties of the transition kernel used. We show how Fill's version of rejection sampling can be extended to produce perfect samples of the penetrable spheres mixture process and related models. Following Häggström et al. we use Gibbs sampling and make use of the partial order of the mixture model state space. Thus we construct an algorithm which protects agains bias caused by user impatience and which delivers samples not only of the mixture model but also of the attractive area-interaction and the continuum random-cluster process. James Propp and David Wilson. Coupling from the past: a user's guide. In D. Aldous and J. Propp, editors, Microsurveys in Discrete Probability , volume 41 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science , pages 181--192. American Mathematical Society, 1998. (Postscript)
An expository paper describing a variety of very different situations for which coupling-from-the-past may be applied, and gives advice on recognizing additional applications. Includes some caveats for the practitioner. Olle Häggström and Karin Nelander. On exact simulation of Markov random fields using coupling from the past. Scandinavian Journal of Statistics , 26(3):395--411, 1999.
Abstract: A general framework for exact simulation of Markov random fields using the Propp-Wilson coupling from the past approach is proposed. Our emphasis is on situations lacking the monotonicity properties that have been exploited in previous studies. A critical aspect is the convergence time of the algorithm; this we study both theoretically and experimentally. Our main theoretical result in this direction says, roughly, that if interactions are sufficiently weak, then the expected running time of a carefully designed implementation is O(N log N) , where N is the number of interacting components of the system. Computer experiments are carried out for random q - colourings and for the Widom-Rowlinson lattice gas model.
Abstract: We study the existence of finitary codings (also called finitary homomorphisms or finitary factor maps) from a finite-valued i. i.d. process to certain random fields. For Markov random fields we show, using ideas of Marton and Shields, that the presence of a phase transition is an obstruction for the existence of the above coding: this yields a large class of Bernoulli shifts for which no such coding exists. Conversely, we show that for the stationary distribution of a monotone exponentially ergodic probabilistic cellular automaton such a coding does exist. The construction of the coding is partially inspired by the Propp-Wilson algorithm for exact simulation. In particular, combining our results with a theorem of Martinelli and Olivieri, we obtain the fact that for the plus state for the ferromagnetic Ising model on Z d , d ≥2, there is (not) such a coding when the interaction parameter is below (above) its critical value. J. N. Corcoran and R. L. Tweedie. Perfect sampling of ergodic Harris chains. Annals of Applied Probability 11(2):438--451, 2001. (Postscript.)
Abstract: We develop an algorithm for simulating ``perfect'' random samples from the invariant measure of a Harris recurrent Markov chain. The method uses backward coupling of embedded regeneration times, and works most effectively for finite chains, or on continuous spaces for stochastically monotone chains, where paths may be sandwiched between ``upper'' and ``lower'' processes. Examples show that more naive approaches to constructing such bounding processes may be considerably biased, but that the algorithm can be simplified in certain cases to make it easier to run. We give explicit analytic bounds on the backward coupling times in the stochastically monotone case. An application of the algorithm to bounded storage models is given, and we also develop a random ``upper'' process for analyzing unbounded storage models.
Abstract: This reports gives a review of the new exact simulation algorithms using Markov chains. The first part covers the discrete case. We consider two different algorithms, Propp and Wilsons ``coupling from the past'' (CFTP) technique and Fills rejection sampler. The algorithms are tested on the Ising model, with and without an external field. The second part covers continuous state spaces. We present several algorithms developed by Murdoch and Green, all based on ``coupling from the past''. We discuss the applicability of these methods on a Bayesian analysis problem of surgical failure rates. Peter J. Green and Duncan J. Murdoch. Exact sampling for Bayesian inference: towards general purpose algorithms (with discussion). In J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, editors, Bayesian Statistics 6 , pages 301--321. Oxford University Press, 1999. Presented as an invited paper at the 6th Valencia International Meeting on Bayesian Statistics, Alcossebre, Spain, June 1998. Preprint.
Abstract: Propp and Wilson (1996) described a protocol, called coupling from the past , for exact sampling from a target distribution using a coupled Markov chain Monte Carlo algorithm. In this paper we discuss the implementation of coupling from the past for samplers on a continuous state space; our ultimate objective is Bayesian MCMC with guaranteed convergence. We make some progress towards this objective, but our methods are still not automatic or generally applicable.
Per author's description: Describes several coupling techniques aimed at routine application in the context of Bayesian inference using random walk Metropolis: the ``bisection coupler'', calculation of automatic cell bounds and the adaptation of the Kendall (1996)/Haggstrom et al (1996) idea of bounded CFTP. Includes simulations of the posterior in two 2-parameter blood type datasets with a Dirichlet prior, a 3 dimensional Dirichlet distribution, and a N(0,1) distribution. W. S. Kendall and Elke Thönnes. Perfect Simulation in Stochastic Geometry. Pattern Recognition 32(9):1569--1586, 1999. Special issue on random sets. (Postscript.)
Abstract: Simulation plays an important rфle in stochastic geometry and related fields, because all but the simplest random set models tend to be intractable to analysis. Many simulation algorithms deliver (approximate) samples of such random set models, for example by simulating the equilibrium distribution of a Markov chain such as a spatial birth-and-death process. The samples usually fail to be exact because the algorithm simulates the Markov chain for a long but finite time, and thus convergence to equilibrium is only approximate. The seminal work by Propp and Wilson made an important contribution to simulation by proposing a coupling method, Coupling from the Past (CFTP), which delivers perfect, that is to say exact, simulations of Markov chains. In this paper we introduce this new idea of perfect simulation and illustrate it using two common models in stochastic geometry: the dead leaves model and a Boolean model conditioned to cover a finite set of points.
Abstract: ``Perfect sampling'' enables exact draws from the invariant measure of a Markov chain. We show that the independent Metropolis-Hastings chain has certain stochastic monotonicity properties that enable a perfect sampling algorithm to be implemented, at least when the candidate is overdispersed with respect to the target distribution. We prove that the algorithm has an optimal geometric convergence rate, and applications show that it may converge extremely rapidly. Persi Diaconis and David Freedman. Iterated random functions. SIAM Review , 41(1):45--76, 1999. (Postscript, PDF.)
Abstract: Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average. The Editors. Graphical illustration of some examples related to the article ``Iterated random functions'' by Diaconis and Freedman. SIAM Review , 41(1):77--82, 1999.
Abstract: Figures are presented to illustrate some examples of iterated random functions of the kind described in the article in this issue by Diaconis and Freedman. Jesper Møller and Katja Schladitz. Extensions of Fill's algorithm for perfect simulation. Journal of the Royal Statistical Society, Ser. B , 61(4):955--969, 1999. (Postscript.)
Abstract: Fill's algorithm for perfect simulation for attractive finite state space models, unbiased for user impatience, is presented in terms of stochastic recursive sequences and extended in two ways. Repulsive discrete Markov random fields with two coding sets like the auto-Poisson distribution on a lattice with 4-neighbourhood can be treated as monotone systems if a particular partial ordering and quasimaximal and quasi-minimal states are used. Fill's algorithm then applies directly. Combining Fill's rejection sampling with sandwiching leads to a version of the algorithm, which works for general discrete conditionally specified repulsive models. Extensions to other types of models are briefly discussed. Mark Huber. Exact Sampling and Approximate Counting Techniques. In Proceedings of the 30th ACM Symposium on the Theory of Computing , pages 31--40, 1998. (Postscript) (PDF)
Per author's description: This paper introduces the idea of a bounding chain for determining when complete coupling has occurred so that Coupling From the Past may be run on the chain to obtain exact samples. A bounding chain for the antiferromagnetic Potts model is given which is shown to couple completely in polynomial time given either a large number of colors or a sufficiently high temperature. The same analytical techniques also show the single-site update model of Propp and Wilson (ferromagnetic Ising) and Häggström and Nelander (antiferromagnetic Ising) run in polynomial time given high enough temperature.
Some of the techniques in this paper were independently developed by Häggström and Nelander. Karin Nelander. A Markov chain Monte Carlo study of the beach model. Markov Processes and Related Fields 6(3):345--369, 2000.
Abstract: We study the beach model introduced by Burton and Steif. The model is an example of a strongly irreducible subshift of finite type which, when the parameter of the model exceeds a critical value, has more than one measure of maximal entropy. We are interested in the critical value and how it depends on dimension. By way of simulations, we find that the critical value seems to be decreasing as a function of the dimension. We also present a conjecture for the whereabouts of the critical value in 2 dimensions. Our simulations are made with Markov chain Monte Carlo methods. For some parameter values and sizes of systems we are able to use the Propp-Wilson algorithm for exact simulation. For other combinations of parameter value and size we use a variant of the Swendsen-Wang algorithm. Krzysztof Burdzy and Wilfrid S. Kendall. Efficient Markovian couplings: examples and counterexamples. The Annals of Applied Probability 10(2):362--409, 2000. (Postscript)
Abstract: In this paper we study the notion of an efficient coupling of Markov processes. Informally, an efficient coupling is one which couples at the maximum possible exponential rate, as given by the spectral gap. This notion is of interest not only for its own sake, but also of growing importance arising from the very recent advent of methods of perfect simulation: it helps to establish the price of perfection' for such methods. In general one can always achieve efficient coupling if the coupling is allowed to cheat (if each component's behaviour is affected by future behaviour of the other component), but the situation is more interesting if the coupling is required to be co-adapted. We present an informal heuristic for the existence of an efficient coupling, and justify the heuristic by proving rigorous results and examples in the contexts of finite reversible Markov chains and of reflecting Brownian motion in planar domains. Mark Huber. A bounding chain for Swendsen-Wang. Random Structure and Algorithms 22(1):43--59, 2003. (PDF)
A two-page version appeared in Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (1999) .
Abstract: The greatest drawback of Monte Carlo Markov chain methods is lack of knowledge of the mixing time of the chain. The use of bounding chains solves this difficulty for some chains by giving theoretical and experimental upper bounds on the mixing time. Moreover, when used with methodologies such as coupling from the past, bounding chains allow the user to take samples drawn exactly from the stationary distribution without knowledge of the mixing time. Here we present a bounding chain for the Swendsen-Wang process. The Swendsen-Wang bounding chain allow us to efficiently obtain exact samples from the ferromagnetic Q-state Potts model for certain classes of graphs. Also, by analyzing this bounding chain, we will show that Swendsen-Wang is rapidly mixing over a slightly larger range of parameters than was known previously. Duncan J. Murdoch and Jeffrey S. Rosenthal. Efficient use of exact samples. Statistics and Computing 10(3):237--243, 2000.
Abstract: Propp and Wilson (1996,1998) described a protocol called coupling from the past (CFTP) for exact sampling from the steady-state distribution of a Markov chain Monte Carlo (MCMC) process. In it a past time is identified from which the paths of coupled Markov chains starting at every possible state would have coalesced into a single value by the present time; this value is then a sample from the steady-state distribution. Unfortunately, producing an exact sample typically requires a large computational effort. We consider the question of how to make efficient use of the sample values that are generated. In particular, we make use of regeneration events (cf. Mykland et al., 1995) to aid in the analysis of MCMC runs. In a regeneration event, the chain is in a fixed reference distribution; this allows the chain to be broken up into a series of tours which are independent, or nearly so (though they do not represent draws from the true stationary distribution). In this paper we consider using the CFTP and related algorithms to create tours. In some cases their elements are exactly in the stationary distribution; their length may be fixed or random. This allows us to combine the precision of exact sampling with the efficiency of using entire tours. Several algorithms and estimators are proposed and analysed. James Allen Fill, Motoya Machida, Duncan J. Murdoch, and Jeffrey S. Rosenthal. Extension of Fill's perfect rejection sampling algorithm to general chains. Extended abstract to appear in Neil Madras, editor, Monte Carlo Methods , volume 26 of Fields Institute Communications , pages 37--52. American Mathematical Society, 2000. Full version appears in a special double issue of Random Structures and Algorithms 17(3&4):290--316, 2000.
Abstract: By developing and applying a broad framework for rejection sampling using auxiliary randomness, we provide an extension of the perfect sampling algorithm of Fill (1998) to general chains on quite general state spaces, and describe how use of bounding processes can ease computational burden. Along the way, we unearth a simple connection between the Coupling From The Past (CFTP) algorithm originated by Propp and Wilson (1996) and our extension of Fill's algorithm.
Remarks: The two versions of the four-authored paper supersede the 1998 preprint.
Duncan J. Murdoch and Jeffrey S. Rosenthal. ``An extension of Fill's exact sampling algorithm to non-monotone chains.'' Olle Häggström and Jeffrey E. Steif. Propp-Wilson algorithms and finitary codings for high noise Markov random fields. Combinatorics, Probability and Computing 9(5):425--439, 2000. (Postscript.)
Abstract: In this paper, we combine two previous works, the first being by the first author and K. Nelander, and the second by J. van den Berg and the second author, to show (1) that one can carry out a Propp--Wilson exact simulation for all Markov random fields on Z d satisfying a certain high noise assumption, and (2) that all such random fields are a finitary image of a finite state i. i.d. process. (2) is a strengthening of the previously known fact that such random fields are Bernoulli shifts. James P. Hobert, Christian P. Robert, and D. M. Titterington. On perfect simulation for some mixtures of distributions. Statistics and Computing 9(4):287--298, 1999. (Postscript.)
Abstract: This paper studies the implementation of the coupling from the past (CFTP) method of Propp and Wilson (1996) in the set-up of two and three component mixtures with known components. We show that monotonicity structures can be exhibited in both cases, but that CFTP can still be costly for three component mixtures. We conclude with a simulation experiment exhibiting an almost perfect sampling scheme where we only consider a subset of the exhaustive set of starting values. C. C. Holmes and B. K. Mallick. Perfect simulation for orthogonal model mixing, 1998. Preprint.
Abstract: In this article we demonstrate how to generate independent and identically distributed samples from the model space of the Bayes linear model with orthogonal predictors. We use the method of coupled Markov chains from the past as introduced by Propp and Wilson (1996). This procedure alleviates any concerns over convergence and sample mixing. We present a number of examples including a perfect simulation of Bayesian wavelet selection in a 1024 dimensional model space, a knot selection problem for spline smoothers and, a standard linear regression variable selection problem.
Abstract: Perfect sampling allows exact simulation of random variables from the stationary measure of a Markov chain. By exploiting monotonicity properties of the slice sampler we show that a perfect version of the algorithm can be easily implemented, at least when the target distribution is bounded. Various extensions, including perfect product slice samplers, and examples of applications are discussed. Christian P. Robert, editor. Discretization and MCMC Convergence Assessment . Lecture Notes in Statistics # 135. Springer, 1998.
Chapter 1, ``Markov chain Monte Carlo methods'' by Christian P. Robert and Sylvia Richardson, has a section 4 on ``Perfect sampling.''
Abstract: We explore and relate two notions of monotonicity, stochastic and realizable, for a system of probability measures on a common finite partially ordered set (poset) S when the measures are indexed by another poset A . We give counterexamples to show that the two notions are not always equivalent, but for various large classes of S we also present conditions on the poset A that are necessary and sufficient for equivalence. When A = S , the condition that the cover graph of S have no cycles is necessary and sufficient for equivalence. This case arises in comparing applicability of the perfect sampling algorithms of Propp and Wilson and the first author of the present paper. Mark Huber. Exact random sampling from independent sets, 1998. Preprint.
Abstract: We consider the problem of randomly sampling from the set of indpendent sets of a graph, where the probability of selecting a particular independent set is proportional to λ raised to the size of the independent set. This distribution is the hard core gas model in statistical physics. If Δ is the maximum degree of the graph, Dyer and Greenhill gave a method for efficiently approximately sampling from this distribution when λ≤2/(Δ-2) by utilizing a rapidly mixing Markov chain. We show here how to use their chain to exactly sample from this distribution. This method is shown to run slightly faster than the method of Dyer and Greenhill, and as a corollary gives a better bound on the mixing time of their chain. Mark Huber. Interruptible exact sampling and construction of strong stationary times for Markov chains, 1998. Preprint.
Abstract: We consider the problem of exactly sampling from the stationary distribution of a Markov chain, a problem with numerous Monte Carlo applications. The earliest method for accomplishing this was construction of a strong stationary times, a stopping time that when reached, left the chain in the stationary distribution. Unfortunately for many chains of interest, these stopping times were very difficult to compute. Here we show how an algorithm of Murdoch and Rosenthal leads to construction of strong stationary times for many chains of practical interest. In addition, we analyze the running time of their algorithm, and compare to the running time of another successful algorithm for exact sampling, coupling from the past. We show how to test whether these stopping times have been reached for several chains for distributions of interest, the hard core gas model, proper k colorings of a graph, and the continuous Widom-Rowlinson mixture model. Armin S. A. Rährl. Fast, portable, parallel and scalable exact simulation using Markov chains, 1999. Preprint.
Abstract: Markov Chain Monte Carlo samplers have become popular tools for Bayesian computation. However, these techniques are computationally expensive. In order to speed them up, this paper presents portable, parallel and linear scalable computer implementations in the Bulk Synchronous Parallel model and in Java. Optimality in scalability and asymptotic optimality in speed is proved. It is shown how this parallel approach can be used for exact sampling using Propp and Wilson's algorithm. In 40 seconds it performs more than one million iterations using a 8192 X 8192 transition matrix of double precision floating point numbers on a SGI with 32 R10000 processors. For time-critical computational applications, an efficient use of the sample values generated via a repeated coupling from the past algorithm is made. Therefore many more samples per time are obtained, though these samples are not drawn from the true stationary distribution. How to develop and analyze a parallel solution in two rather different models of computation is explained. Reference to the complete code with documentation is provided in [20]. Wilfrid S. Kendall and Jesper Møller. Perfect implementation of a Metropolis-Hastings simulation of Markov point processes, 1999. Preprint.
Abstract: This document describes a C implementation of a perfect algorithm for a Metropolis-Hastings simulation of a point process, based on a decomposition into types relating to different square sub-regions, and a dominating process which is composed of independent discrete-time M/M/1) queues, one for each square sub-region, each in equilibrium. Mathematical details are to be found in Kendall & Møller (1999a), which is the scientific paper corresponding to this research report. The differences between the current document and Kendall & Møller (1999a) are that this document gives a full listing of the program, this document does not discuss the underlying theory, which is covered in depth in Kendall & Møller (1999a).
A full Nuweb source for this program (in gzipped tarfile form, together with figures) is available at my [Kendall's] [homepage of available perfect simulation programs].
A system (P a : a in A) of probability measures on a common finite set S indexed by another finite set A can be "realized" by a system (X a : a in A) of S - valued random variables on some probability space so that X a is distributed as P a for each a in A . Assuming that A and S are each equipped with a partial order, we pose the following question: When can a given system (P a : a in A) be realized by a system (X a : a in A) with the monotonicity property that X a ≤ X b almost surely whenever a Jesper Møller and G. K. Nicholls. Perfect simulation for sample-based inference. Statistics and Computing, to appear. (PDF)
Abstract: Perfect simulation algorithms based on Propp and Wilson (1996) have so far been of limited use for sampling problems of interest in statistics. We specify a new family of perfect sampling algorithms obtained by combining MCMC tempering algorithms with dominated coupling from the past, and demonstrate that our algorithms will be useful for sample based inference. Perfect tempering algorithms are less efficient than the MCMC algorithms on which they typically depend. However, samples returned by perfect tempering are distributed according to the intended distribution, so that these new sampling algorithms do not suffer from the convergence problems of MCMC. Perfect tempering is related to rejection sampling. When rejection sampling has been tried, but has proved impractical, it may be possible to convert the rejection algorithm into a perfect tempering algorithm, with a significant gain in algorithm efficiency. Mark Huber. The swap move: a tool for building faster Markov chains, 1999. Preprint.
Abstract: In Monte Carlo Markov chain algorithms, a random sample is generated by running a Markov chain ``for a long time''. This technique has applications in many areas, including statistical physics and approximation algorithms. Two questions about this procedure arise, first, how long does the chain need to be run before the sample obtained is close to being in the desired distribution, and secondly, how should chains be designed to make the running time as low as possible? Here we present a method for increasing the speed of Markov chains that we call the swap move. First applied by Broder to approximate the permanent, here we apply it to several models from statistical physics. In each case we are able to create improved Markov chains for the problem. More importantly, we show how the concept of bounding chains may be applied to the swap move, thus allowing the use of perfect sampling algorithms such as coupling from the past. These algorithms eliminate the need to know the mixing time of the Markov chain, making these new chains always faster to use.
Mark Huber. Perfect sampling without a lifetime commitment, 1999. Preprint.
Abstract: Generating perfect samples from distributions using Markov chains has a wide range of applications, from statistical physics to approximation algorithms. In perfect sampling algorithms, a sample is drawn exactly from the stationary distribution of a chain, as opposed to methods that run the chain ``for a long time'' and create samples drawn from a distribution that is close to the stationary distribution. One of the primary methods for creating perfect samples, the coupling from the past protocol of Propp and Wilson, suffers from the fact that the user must be willing to commit to running the algorithm in its entirety in order to obtain unbiased, perfect samples. By using another method, the acceptance rejection method of Fill, Murdoch, and Rosenthal (FMR), the user may abort the procedure in the middle of a run without introducing bias. In this paper, we give the first general analysis of the running time of FMR compared to the running time of of CFTP. Furthermore, we show that FMR can be used to generate a strong stationary stopping time of a Markov chain. We show how to use FMR to generate perfect sampling algorithms for two real world examples, the hard core gas model and the Widom-Rowlinson mixture model. These two examples illustrate the techniques needed to apply FMR to a wide range of discrete and infinite state space chains.
Remarks: Two references for the general form of Fill's algorithm are (1) ``Fill'' (his original article) and (2) ``Fill, Machida, Murdoch, and Rosenthal'' (which supersedes the preprint by Murdoch and Rosenthal).
Abstract: We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related to an idea known as PASTA (Poisson arrivals see time averages) in the operations research literature. Because the new algorithm can be run using a read-once stream of randomness, we call it read-once CFTP. The memory and time requirements of read-once CFTP are on par with the requirements of the usual form of CFTP, and for a variety of applications the requirements may be noticeably less. Some perfect sampling algorithms for point processes are based on an extension of CFTP known as coupling into and from the past; for completeness, we give a read-once version of coupling into and from the past, but it remains unpractical. For these point process applications, we give an alternative coupling method with which read-once CFTP may be efficiently used. Xiao-Li Meng. Towards a more general Propp-Wilson algorithm: multistage backward coupling. In Neil Madras, editor, Monte Carlo Methods , volume 26 of Fields Institute Communications . American Mathematical Society, 2000. To appear.
Abstract: A multistage implementation of Propp and Wilson's backward coupling scheme is proposed via cluster sampling, which can help reduce the time to coalescence. A random walk example is used to illustrate the multistage generalization and to compare it with Propp and Wilson's single-stage scheme. The use of backward cluster sampling is also proposed for stratified forward Markov chain Monte Carlo, taking advantage of the greater flexibility of the multistage framework. These proposals were inspired by a general principle in sample surveys --- multistage sampling is typically more effective than single stage sampling. George Casella, Kerrie L. Mengersen, Christian P. Robert and D. M. Titterington). Perfect slice samplers for mixtures of distributions. Journal of the Royal Statistical Society, Ser. B 64(4):777--790, 2002. (Postscript.)
Abstract: We propose a perfect sampler for mixtures of distributions, in the spirit of Mira and Roberts (1999), building on Hobert, Robert and Titterington (1999). The method relies on a marginalisation akin to Rao-Blackwellisation which illustrates the Duality Principle of Diebolt and Robert (1994) and utilises an envelope argument which embeds the finite support distribution on the latent variables within a continuous support distribution, easier to simulate by slice sampling. We also provide a number of illustrations in the cases of normal and exponential mixtures which show that the technique does not suffer from severe slow-down when the number of observations or the number of components increases. We thus obtain a general iid sampling method for mixture posterior distributions and illustrate convincingly that perfect sampling can be achieved for realistic statistical models and not only for toy problems.
Abstract: We discuss the interrelation between phase transitions in interacting lattice or continuum models, and the existence of infinite clusters in suitable random-graph models. In particular, we describe a random-geometric approach to the phase transition in the continuum Ising model of two species of particles with soft or hard interspecies repulsion. We comment also on the related area-interaction process and on perfect simulation. D. J. Murdoch. Exact sampling for Bayesian inference: unbounded state spaces. In Neil Madras, editor, Monte Carlo Methods , volume 26 of Fields Institute Communications . American Mathematical Society, 2000. To appear. (Postscript)
Abstract: Propp and Wilson [10, 11] described a protocol called coupling from the past (CFTP) for exact sampling from the steady-state distribution of a Markov chain Monte Carlo (MCMC) process. In it a past time is identified from which the paths of coupled Markov chains starting at every possible state would have coalesced into a single value by the present time; this value is then a sample from the steady-state distribution.
Foss and Tweedie [3] pointed out that for CFTP to work, the underlying Markov chain must be uniformly ergodic. Unfortunately, most of the chains in common use in Bayesian inference are not, when the state space is unbounded. However, this does not mean that CFTP can't be used; in this paper we present three modifications.
The first is a simple change to the chain to induce uniform ergodicity. The second (more extensively discussed in [4]) is a modification to CFTP due to Kendall [6] that makes use of random bounds on particular realizations of the chain. Finally, the last method attempts to make use of Meng's [7] multistage coupler to address the problem. David B. Wilson. Layered multishift coupling for use in perfect sampling algorithms (with a primer on CFTP). In Neil Madras, editor, Monte Carlo Methods , volume 26 of Fields Institute Communications , pages 141--176. American Mathematical Society, 2000. arXiv:math. PR/9912225.
Abstract: In this article we describe a new coupling technique which is useful in a variety of perfect sampling algorithms. A multishift coupler generates a random function f() so that for each real x , f(x)-x is governed by the same fixed probability distribution, such as a normal distribution. We develop the class of layered multishift couplers, which are simple and have several useful properties. For the standard normal distribution, for instance, the layered multishift coupler generates an f() which (surprisingly) maps an interval of length l to fewer than 2+l/2.35 points --- useful in applications which perform computations on each such image point. The layered multishift coupler improves and simplifies algorithms for generating perfectly random samples from several distributions, including and autogamma distribution, posterior distributions for Bayesian inference, and the steady state distribution for certain storage systems. We also use the layered multishift coupler to develop a Markov-chain based perfect sampling algorithm for the autonormal distribution.
At the request of the organizers, we begin by giving a primer on CFTP (coupling from the past); CFTP and Fill's algorithm are the two predominant techniques for generating perfectly random samples using coupled Markov chains. Haiyan Cai. Exact sampling using auxiliary variables, 1999. To Appear in the Statistical Computation Section of the ASA Proceedings. (Postscript.)
Abstract: We discuss two simple algorithms for generating exact samples from the equilibrium distribution of a Markov chain. These algorithms are based on a single run of the Markov chain in an ordinary way. One can pick up certain samples from the run as samples from the equilibrium distribution. This is done by introducing auxiliary variables as the indicators during the run. Alessandra Guglielmi, Chris C. Holmes, and Stephen G. Walker. Perfect simulation involving functionals of a Dirichlet process. Journal of Computational and Graphical Statistics 11(2):306--310, 2002. (Postscript)
Abstract: In this paper, we show how to perform perfect simulation of a functional of a Dirichlet process, when the values of the functional are bounded. We also give a procedure for ``approximate'' simulation in the case of unbounded functional values. Xeni K. Dimakos. A guide to exact simulation. International Statistical Review 69(1):27--48, 2001. (Postscript)
Abstract: Markov Chain Monte Carlo (MCMC) methods are used to sample from complicated multivariate distributions with normalizing constants that may not be computable and from which direct sampling is not feasible. A fundamental problem is to determine convergence of the chains. Propp & Wilson (1996) devised a Markov chain algorithm called Coupling From The Past (CFTP) that solves this problem, as it produces exact samples from the target distribution and determines automatically how long it needs to run. Exact sampling by CFTP and other methods is currently a thriving research topic. This paper gives a review of some of these ideas, with emphasis on the CFTP algorithm. The concepts of coupling and monotone CFTP are introduced, and results on the running time of the algorithm presented. The interruptible method of Fill (1998) and the method of Murdoch & Green (1998) for exact sampling for continuous distributions are presented. Novel simulation experiments are reported for exact sampling from the Ising model in the setting of Bayesian image restoration, and the results are compared to standard MCMC. The results show that CFTP works at least as well as standard MCMC, with convergence monitored by the method of Raftery & Lewis (1992, 1996). M. A. Novotny. Introduction to the Propp-Wilson method of exact sampling for the Ising model. In D. P. Landau, S. P. Lewis, and H.-B. Schьttler, editors, Computer Simulation Studies in Condensed Matter Physics XII , volume 85 of Springer Proceedings in Physics , pages 179--184. Springer Verlag, 2000. arXiv:cond-mat/9905195.
Abstract: An introduction to the Propp-Wilson method of coupling-from-the-past for the Ising model is presented. It enables one to obtain exact samples from the equilibrium spin distribution for ferromagnetic interactions. Both uniform and random quenched magnetic fields are included. The time to couple from the past and its standard deviation are shown as functions of system size, temperature, and the field strength. The time should be an estimate of the ergodic time for the Ising system. Elke Thönnes. A primer on perfect simulation. Statistical Physics and Spatial Statistic, ed. Klaus R. Mecke and D. Stoyan, Springer Lecture Notes in Physics #554, 349--378, 2000.
Abstract: Markov chain Monte Carlo has long become a very useful, established tool in statistical physics and spatial statistics. Recent years have seen the development of a new, exciting generation of Markov chain Monte Carlo methods: perfect simulation algorithms. In contrast to conventional Markov chain Monte Carlo perfect simulation produces samples which are guaranteed to have the exact equilibrium distribution. In the following we provide an example-based introduction into perfect simulation focussed on the method called Coupling from the Past. Michael Harvey. Monte Carlo inference for belief networks using coupling from the past. Master's thesis, department of computer science, University of Toronto, 1999. (Postscript, PDF.)
Abstract: A common method of inference for belief networks is Gibbs sampling, in which a Markov chain converging to the desired distribution is simulated. In practice, however, the distribution obtained with Gibbs sampling differs from the desired distribution by an unknown error, since the simulation time is finite. Coupling from the past selects states from exactly the desired distribution by starting chains in every state at a time far enough back in the past that they reach the same state at time t = 0. To track every chain is an intractable procedure for large state spaces. The method proposed in this thesis uses a summary chain to approximate the set of chains. Transitions of the summary chain are efficient for noisy-or belief networks, provided that sibling variables of the network are not directly connected, but often require more simulation time steps than would be needed if chains were tracked exactly. Testing shows that the method is a potential alternative to ordinary Gibbs sampling, especially for networks that have poor Gibbs sampling convergence, and when the user has a low error tolerance. Tine Møller Sørensen. A study of Propp & Wilson's method for exact Markov chain Monte Carlo sampling, 1997. Master's thesis in computational statistics, University of Bath.
Abstract: The aim of this project is to study the Propp & Wilson method (Propp & Wilson, 1996) for exact simulation via Markov chain Monte Carlo.
The first part of the project involves investigating the computation time required for Propp & Wilson's algorithm in different applications. The objective is to find the optimal division of effort between `pre-sampling' which ensures the Markov chain is sampling from its ergodic distribution and the subsequent `production runs' which yield internally correlated realisations from which to draw inferences about the target distribution. In our application, it is found that the most accurate estimates are obtained by using one long run of correlated samples instead of several shorter runs, although there are other advantages of using several runs.
In the second part of the project different sampling methods based on the Propp & Wilson algorithm are studied in situations where exact sampling is not feasible. The objective of this part of the project is to develop methods that can be used as diagnostics for exact sampling in these cases. We consider a number of distributions and according to the situation, different methods are tried out. Furthermore, a test is developed to assess the methods. Roberto Fernández, Pablo A. Ferrari, and Nancy L. Garcia. Perfect simulation for interacting point processes, loss networks and Ising models, 1999. math. PR/9911162.
Abstract: We present a perfect simulation algorithm for measures that are absolutely continuous with respect to some Poisson process and can be obtained as invariant measures of birth-and-death processes. Examples include area - and perimeter-interacting point processes (with stochastic grains), invariant measures of loss networks, and the Ising contour and random cluster models. The algorithm does not involve couplings of the process with different initial conditions and it is not tied up to monotonicity requirements. Furthermore, it directly provides perfect samples of finite windows of the infinite-volume measure, subjected to time and space ``user-impatience bias''. The algorithm is based on a two-step procedure: (i) a perfect-simulation scheme for a (finite and random) relevant portion of a (space-time) marked Poisson processes (free birth-and-death process, free loss networks), and (ii) a ``cleaning'' algorithm that trims out this process according to the interaction rules of the target process. The first step involves the perfect generation of ``ancestors'' of a given object, that is of predecessors that may have an influence on the birth-rate under the target process. The second step, and hence the whole procedure, is feasible if these ``ancestors'' form a finite set with probability one. We present a sufficiency criteria for this condition, based on the absence of infinite clusters for an associated (backwards) oriented percolation model. Bas Straatman. Exact sampling and applications to a mite dispersal model, 1998. Master's thesis, Utrecht.
This thesis does not itself come with an abstract, but the abstract for a talk given by Straatman on the topic follows.
Abstract: I have studied a Markov chain which describes the dispersal of mites on cassava fields in West Africa. Instead of obtaining a sample of this model with approximately the stationary distribution by Monte Carlo simulation, it is possible to obtain a result which is exactly distributed according to the stationary distribution. One way to do this is coupling from the past. After a short introduction of this procedure I will explain the changes I have made to the original model in order to be able to use coupling from the past and show some simulations which give exact samples. Jens Lund and Elke Thönnes. Perfect simulation and inference for point processes given noisy observations. Computational Statistics 19(2):317--336, 2004. ( Postscript)
Abstract: The paper is concerned with the Bayesian analysis of point processes which are observed with noise. It is shown how to produce exact samples from the posterior distribution of the unobserved true point process given a noisy observation. The algorithm is a perfect simulation method which applies dominated coupling from the past to a spatial birth-and-death process. Dominated coupling from the past is made amenable by augmenting the state space of the target chain. We discuss how to optimize this perfect simulation algorithm and how to use it in a statistical inference problem of pratical importance. We further describe the merits and the limitations of the method. Jens Lund. Statistical inference and perfect simulation for point processes observed with noise . PhD thesis, The Royal Veterinary and Agricultural University, Copenhagen, Department of Statistics, 1999.
Abstract: The main themes of this thesis are spatial statistics and simulation algorithms. The thesis is split into five papers that may be read independently. All five papers deal with spatial models. Lund & Rudemo (1999), Lund et al. (1999), and Lund & Thönnes (1999b) deal with the same new model for point processes observed with noise, and Lund et al. (1999), Lund & Thönnes (1999b), and Lund & Thönnes (1999a) has a simulation aspect.
Lund & Rudemo (1999), Lund et al. (1999), and Lund & Thönnes (1999b) develop and analyse a new model for point processes observed with noise. Usually the analysis of spatial point patterns assume that the observed points (the true points) are a realization from a specific model. In contrast our approach is to assume the observed pattern generated by thinning and displacement of the true points, and allow for contamination by points not belonging to the true pattern.
Lund & Rudemo (1999) develop the model for point processes observed with noise. The likelihood function for an observation of a noise corrupted point pattern given the true positions is derived. As data for our analysis is indeed a realization of the underlying true process and its associated noise corrupted point pattern we need not consider a model for the underlying process. The parameters in the model describe how many of the true points are lost, how large the displacements are, and the number of contaminating surplus points. For estimation of the parameters in the noise model a deterministic, iterative, and approximative maximum likelihood estimation algorithm is developed. The likelihood function is a sum of an excessive large number of terms, and the algorithm works by finding large dominating terms. Alternative estimation methods are discussed.
Lund et al. (1999) analyse the model developed in Lund & Rudemo (1999) with respect to the now unobserved true points. We assume a noisy observation of a true point pattern and knowledge of the parameters in the model. A Bayesian point of view is now adopted and we specify a prior distribution for the underlying true process. Given the model, the prior distribution, and the noisy observation, we get the posterior distribution of the true points. This posterior distribution is investigated by samples from the distribution. These samples are obtained from a Markov chain Monte Carlo (MCMC) algorithm extending the Metropolis-Hastings sampler for point processes. A thorough discussion is provided on the choice of prior distribution and how to present the samples from the MCMC runs. The MCMC samples are used to estimate for example the K - function for the unobserved true point pattern. These estimates are clearly better than estimates based on the observed points alone.
The use of the MCMC algorithm in Lund et al. (1999) relies on the fact that a Markov chain run for a long time approaches its stationary distribution. Lund & Thönnes (1999b) uses a recent technique called Coupling From The Past (CFTP) to deliver a sample drawn from the exact posterior distribution of the unobserved true points described in Lund et al. (1999), a so-called perfect simulation. This perfect simulation algorithm is based on spatial birth-and-death processes for simulation of point processes. In order to apply CFTP in our problem the simulation is carried out on an augmented state space. The algorithm turns out to be too slow in practice and thus demonstrates possible current limits of CFTP.
Lund & Thönnes (1999a) describes a new perfect simulation algorithm for general locally stable point processes. The algorithm is based on CFTP for Metropolis-Hastings simulation of point processes and it is simpler than the previous known perfect algorithm based on Metropolis-Hastings simulation for this class of models. The present state of the algorithm is that it is far too slow to be useful in practice, and it might have some theoretical flaws. These problems are discussed in the introductory part of the thesis.
Lund (1998) develops a model for survival times of trees that take the spatial positions of the trees into account. At a finite number of timepoints it is observed whether a tree is alive or not, and thus we have interval censoring of the even aged trees. The model is a discrete time version of Cox's proportional hazards model. Positions of trees are considered as fixed, and they are used to compute competition indices that enter the model as covariates. It is shown that small trees have a higher risk of dying than large tress and the area of the experiment is inhomogeneous. In addition, Hegyi's competition index based on basal area is a significant covariate. Alison Gibbs. Bounding the convergence time of the Gibbs sampler in Bayesian image restoration. Biometrika 87(4):749--766, 2000. (Postscript).
Abstract: This paper shows how coupling methodology can be used to give precise, a priori bounds on the convergence time of Markov chain Monte Carlo algorithms for which a partial order exists on the state space which is preserved by the Markov chain transitions. This methodology is applied to give a bound on the convergence time of the random scan Gibbs sampler used in the Bayesian restoration of an image of N pixels. For our algorithm in which only one pixel is updated at each iteration, the bound is a constant times N 2 . The proportionality constant is given and is easily calculated. These bounds also give an indication of the running time of coupling from the past algorithms. Alison Gibbs. Convergence of Markov Chain Monte Carlo Algorithms with Applications to Image Restoration. PhD thesis, University of Toronto, Department of Statistics, 1999. (Postscript, PDF.)
Abstract: Markov chain Monte Carlo algorithms, such as the Gibbs sampler and Metropolis-Hastings algorithm, are widely used in statistics, computer science, chemistry and physics for exploring complicated probability distributions. A critical issue for users of these algorithms is the determination of the number of iterations required so that the result will be approximately a sample from the distribution of interest.
In this thesis, we give precise bounds on the convergence time of the Gibbs sampler used in the Bayesian restoration of a degraded image. We consider convergence as measured by both the usual choice of metric, total variation distance, and the Wasserstein metric. In both cases we exploit the coupling characterisation of the metric to get our results. Our results can also be applied to the coupling-from-the-past algorithm of Propp and Wilson (1996) to get bounds on its running time.
The application of our theoretical results requires the computation of parameters of the algorithm. These computations may be prohibitively difflcult in many situations. We discuss how our results can be applied in these situations through the use of auxiliary simulation to estimate these parameters.
We also give a summary of probability metrics and the relationships between them, including several new relationships. Paul Fearnhead. Perfect simulation from population genetic models with selection. Theoretical Population Biology, 59:263--279, 2001. (Postscript)
Abstract: We consider using the ancestral selection graph to simulate samples from genetic models with selection. Currently the use of the ancestral selection graph to simulate such samples is limited. This is because the computational requirement for simulating such samples increases exponentially with the selection rate, and also due to the need to be able to simulate a sample of size 1 from the population at equilibrium. For the only case where the distribution of a sample of size one is known, that of parent independent mutations, more efficient simulation algorithms exist. We will show that by applying the idea of coupling from the past to the ancestral selection graph, samples can be simulated from a general K - allele model without knowledge of the distribution of a sample of size 1. Further, the computation involved in generating such samples appears to be less than that of simulating the ancestral selection graph until its ultimate ancestor. In particular, in the case of genic selection with parent independent mutations, the computational requirement increases only quadratically with the selection rate. The algorithm is then used to simulate samples at microsatellite loci, and to consider the effect of selection on linked neutral sites. Jesper Møller. Aspects of spatial statistics, stochastic geometry and Markov chain Monte Carlo methods. Thesis for doctoral degree in Natural Sciences, Aalborg University.
From the preface: This doctoral thesis is concerned with certain aspects of spatial statistics, stochastic geometry, and Markov chain Monte Carlo methods. It is based on a monograph [L] and twenty selected papers [A--K, M--V] produced over about the last decade; see the following two pages. The papers [A, G, H, I] and [95, 140] constituted my Ph. D. thesis from 1988 entitled Stochastic Geometry and Markov Models ; [A, H, I] in revised form are included in the present thesis.
The thesis consists of three parts: (i) statistical models for lattice data, spatial (marked) point patterns, and spatial birth-and-death processes, (ii) Markov chain Monte Carlo methods and statistical inference for spatial models, and (iii) random tessellations. Parts (i) and (ii) are closely related and the focus is on recent theoretical and applied statistical aspects. Part (iii) is more within the traditional framework of stochastic geometry, though new results will be discussed as well.
Remark: Section 3.2 deals with perfect simulation. Song-Chun Zhu and David Mumford. Learning Generic Prior Models for Visual Computation, 1997. To appear in IEEE Transactions on Pattern Analysis and Machine Intelligence . (Postscript)
Abstract: Many generic prior models have been widely used in computer vision ranging from image and surface reconstruction to motion analysis, and these models presume that surfaces of objects be smooth, and adjacent pixels in images have similar intensity values. However, there is little rigorous theory to guide the construction and selection of prior models for a given application. Furthermore, images are often observed at arbitrary scales, but none of the existing prior models are scale-invariant. Motivated by these problems, this article chooses general natural images as a domain of application, and proposes a theory for learning prior models from a set of observed natural images. Our theory is based on a maximum entropy principle, and the learned prior models are of Gibbs distributions. A novel information criterion is proposed for model selection by minimizing a Kullback-Leibler information distance. We also investigate scale invariance in the statistics of natural images and study a prior model which has scale invariant property. In this paper, in contrast with all existing prior models, negative potentials in Gibbs distribution are first reported. The learned prior models are verified in two ways. Firstly images are sampled from the prior distributions to demonstrate what typical images they stand for. Secondly they are compared with existing prior models in experiments of image restoration.
Remarks: This paper is about image processing rather than perfect sampling per se , but the end of section 3 includes discussion of their use of the ``Propp-Wilson criterion'' for deciding how long to run their Markov chains. Andrew M. Childs, Ryan B. Patterson, and David J. C. MacKay. Exact sampling from non-attractive distributions using summary states. Physical Review E 63:036113, 2001. arXiv:cond-mat/0005132.
Abstract: Propp and Wilson's method of coupling from the past allows one to efficiently generate exact samples from attractive statistical distributions (e. g., the ferromagnetic Ising model). This method may be generalized to non-attractive distributions by the use of summary states , as first described by Huber. Using this method, we present exact samples from a frustrated antiferromagnetic triangular Ising model and the antiferromagnetic q = 3 Potts model. We discuss the advantages and limitations of the method of summary states for practical sampling, paying particular attention to the slowing down of the algorithm at low temperature. In particular, we show that such a slowing down can occur in the absence of a physical phase transition. Luc Devroye, James A. Fill, and Ralph Neininger. Perfect Simulation from the Quicksort Limit Distribution, 2000. Preprint.
Abstract: The weak limit of the normalized number of comparisons needed by the Quicksort algorithm to sort n randomly permuted items is known to be determined implicitly by a distributional fixed-point equation. We give an algorithm for perfect random variate generation from this distribution. George Casella, Michael Lavine, and Christian P. Robert. Explaining the perfect sampler. The American Statistician 55(4):299--305, 2001. (Postscript).
Abstract: In 1996, Propp and Wilson introduced Coupling from the Past ( CFTP ), an algorithm for generating a sample from the exact stationary distribution of a Markov chain. In 1998, Fill proposed another so-called perfect sampling algorithm. These algorithms have enormous potential in Markov Chain Monte Carlo ( MCMC ) problems because they eliminate the need to monitor convergence and mixing of the chain. This article provides a brief introduction to the algorithms, with an emphasis on understanding rather than technical detail. Olle Häggström. Finite Markov Chains and Algorithmic Applications. Cambridge University Press, 2000.
Contents: These lecture notes give an elementary introduction to Markov chains, and applications to algorithms like MCMC and simulated annealing. Chapters are entitled:
1. Basics of probability theory.
2. Markov chains.
3. Computer simulation of Markov chains.
4. Irreducible and aperiodic Markov chains.
5. Stationary distributions.
6. Reversible Markov chains.
7. Markov chain Monte Carlo.
8. The Propp-Wilson algorithm.
9. Simulated annealing.
10. Further reading. Erik van Zwet. Perfect stochastic EM. In M. C. M. de Gunst, C. A. J. Klaassen, and A. W. van der Vaart, editors, State of the Art in Probability and Statistics , IMS Lecture Notes #36, pages 607--616, 2001. (Postscript.)
Abstract: In a missing data problem we observe the result of a (known) many-to-one mapping of an unobservable `complete' dataset. The aim is to estimate some parameter of the distribution of the complete data. In this situation, the stochastic version of the EM algorithm is sometimes a viable option. It is an iterative algorithm that produces an ergodic Markov chain on the parameter space. The stochastic EM (StEM) estimator is then a sample from the equilibrium distribution of this chain. Recently, a method called `coupling from the past' was invented to generate a Markov chain in equilibrium. We investigate when this method can be used for a StEM chain and give examples where this is indeed possible. W. L. Cooper and R. L. Tweedie. Perfect simulation of an inventory model for perishable products. Stochastic Models , 18(2):229--243, 2002. (Postscript.)
Abstract: We study an inventory model for perishable products with a critical-number ordering policy under the assumption that demand for the product forms an i. i.d. sequence, so that the state of the system forms a Markov chain. Explicit calculation of the stationary distribution has proved impractical in cases where items have reasonably long lifetimes and for systems with large ``under-up-to'' levels. Using the recently-developed coupling-from-the-past method, we introduce a technique to estimate the stationary distribution of the Markov chain via ``perfect simulation.'' The Markov chain that results from the use of a critical-number policy is particularly amenable to these simulation techniques, despite not being ordered in its initial state, since the recursive equations satisfied by the Markov chain enable us to identify specific demand patterns where the backward coupling occurs. Jesper Møller. A review of perfect simulation in stochastic geometry. Selected Proceedings of the Symposium on Inference for Stochastic Processes. Eds. I. V. Basawa, C. C. Heyde and R. L. Taylor, IMS Lecture Notes & Monographs Series, 2001, Volume 37, pages 333--355. (Postscript.)
Abstract: We provide a review and a unified exposition of the Propp-Wilson algorithm and various other algorithms for making perfect simulations with a view to applications in stochastic geometry. Most examples of applications are for spatial point processes. Francis Comets, Roberto Fernández, and Pablo A. Ferrari. Processes with Long Memory: Regenerative Construction and Perfect Simulation, 2000. arXiv:math. PR/0009204.
Abstract: We present a perfect simulation algorithm for stationary processes indexed by Z , with summable memory decay. Depending on the decay, we construct the process on finite or semiinfinite intervals, explicitly from an i. i.d. uniform sequence. Even though the process has infinite memory, its value at time 0 depends only on a finite, but random, number of these uniform variables. The algorithm is based on a recent regenerative construction of these measures by Ferrari, Maass, Martinez and Ney. As applications, we discuss the perfect simulation of binary autoregressions and Markov chains on the unit interval. James A. Fill and Mark Huber. The randomness recycler: a new technique for perfect sampling. 41st Annual Symposium on Foundations of Computer Science , pages 503--511, 2000. arXiv:math. PR/0009242.
Abstract: For many probability distributions of interest, it is quite difficult to obtain samples efficiently. Often, Markov chains are employed to obtain approximately random samples from these distributions. The primary drawback to traditional Markov chain methods is that the mixing time of the chain is usually unknown, which makes it impossible to determine how close the output samples are to having the target distribution. Here we present a new protocol, the randomness recycler (RR), that overcomes this difficulty. Unlike classical Markov chain approaches, an RR-based algorithm creates samples drawn exactly from the desired distribution. Other perfect sampling methods such as coupling from the past use existing Markov chains, but RR does not use the traditionalMarkov chain at all. While by no means universally useful, RR does apply to a wide variety of problems. In restricted instances of certain problems, it gives the first expected linear time algorithms for generating samples. Here we apply RR to self-organizing lists, the Ising model, random independent sets, random colorings, and the random cluster model. James A. Fill and Motoya Machida. Realizable Monotonicity and Inverse Probability Transform, 2000. arXiv:math. PR/0010026.
Abstract: A system (P a : a in A) of probability measures on a common state space S indexed by another index set A can be ``realized'' by a system (X a : a in A) of S - valued random variables on some probability space in such a way that each X a is distributed as P a . Assuming that A and S are both partially ordered, we may ask when the system (P a : a in A) can be realized by a system (X a : a in A) with the monotonicity property that X a Marc A. Loizeaux and Ian W. McKeague. Bayesian inference for spatial point processes via perfect sampling, 2000. Preprint.
Abstract: We study a Bayesian cluster model for spatial point processes in which observations are assumed to cluster around a finite collection of underlying landmarks. Perfect sampling from the posterior distribution of landmarks is investigated. In the case of Neyman-Scott cluster models, perfect sampling from the posterior is shown to be computationally feasible via a coupling-from-the-past type algorithm of Kendall and Møller. An application to data on leukemia incidence in upstate New York is developed. M. N. M. van Lieshout and Erik van Zwet. Maximum likelihood estimation for the bombing model, 2000. Preprint.
Abstract: Perhaps the best known example of a random set is the Boolean model. It is the union of `grains' such as discs, squares or triangles which are placed at the points of a Poisson point process. The Poisson points are called the `germs'. We are interested in estimating the intensity, say lambda, of the Poisson process from a sample of a Boolean model of discs (the bombing model). A natural estimate is the number of germs in the observation region divided by the area of that region. Unfortunately, we do not observe the presence of a given germ when its associated disc is completely covered by other discs. On the other hand, we observe the exact location of a germ when we observe any part of its associated disc's boundary. We demonstrate how to apply Coupling From The Past to sample from the conditional distribution, given the data, of the unobserved germs. Such samples allow us to approximate the maximum likelihood estimator of the intensity. We discuss and compare two methods to do so. The first method is based on a Monte Carlo approximation of the likelihood function. The second is a stochastic version of the EM algorithm. James P. Hobert and Christian P. Robert. Moralizing perfect sampling, 2000. Preprint.
Abstract: Let Φ be an irreducible, aperiodic, Harris recurrent Markov chain with invariant probability measure π. We show that if a minorization condition can be established for Φ, then π can be represented as an infinite mixture from which it may be possible to sample. Making exact draws from this mixture (and hence from π requires the ability to simulate two different random variables, T * and N t . When the small set in the minorization condition is the entire state space (and hence Φ is uniformly ergodic), simulating these random variables is straightforward and the resulting algorithm turns out to be equivalent to Murdoch and Green's (1998) multigamma coupler . Thus, we are able to achieve perfect sampling in the uniformly ergodic case without any appeal to coupling or backward simulation . When the small set is not the state space, simulating T * is more problematic. Fortunately, T * is univariate and has support N . We construct an estimate of the mass function of T * and an asymptotic bound on the error of this estimate. These results are used to address rigorously the issue of burn-in. Specifically, we form an approximation of π whose error can be controlled and which can be used as an initial distribution for Φ. We illustrate this with a realistic example. Finally, we provide a simple Markov chain Monte Carlo (MCMC) algorithm whose stationary distribution is that of T * . Finding a perfect sampler based on our MCMC algorithm for the univariate, discrete T * is tantamount to the ability to sample exactly from π. Michael K. Schneider. Exact and Approximate Sampling from the Stationary Distribution of a Markov Chain, 1998. Area exam report. (postscript, compressed postscript, pdf)
Abstract: When simulating a physical system with discrete sates, one often would like to generate a sample from the stationary distribution of a Markov chain. This report focuses on three sampling methodologies which do not rely on explicitly computing the stationary distribution. Two of these lead to algorithms which can generate an exact sample in finite time. The third yields a sample whose distribution approximates, but is arbitrary close to, the stationary distribution from which one desires a sample. The approximate and one of the exact methodologies are illustrated with examples from statistical mechanics. Krishna B. Athreya and Örjan Stenflo. Perfect Sampling for Doeblin Chains. Sankhya - The Indian Journal of Statistics 65(4):763--777, 2003. (PDF).
Abstract: For Markov chains that can be generated by iteration of i. i.d. random maps from the state space X into itself (this holds if X is Polish) it is shown that the Doeblin minorization condition is necessary and sufflcient for the method by Propp and Wilson for ``perfect'' sampling from the stationary distribution π to be successful.
Using only the transition probability P we produce in a geometrically distributed random number of steps N a ``perfect'' sample from π of size N !. Joakim Linde, Cristopher Moore, Mats G. Nordahl. An n - dimensional generalization of the rhombus tiling, 2001. Preprint.
Abstract: Several classic tilings, including rhombuses and dominoes, possess height functions which allow us to 1) prove ergodicity and polynomial mixing times for Markov chains based on local moves, 2) use coupling from the past to sample perfectly random tilings, 3) map the statistics of random tilings at large scales to physical models of random surfaces, and and 4) are related to the ``arctic circle'' phenomenon. However, few examples are known for which this approach works in three or more dimensions. Here we show that the rhombus tiling can be generalized to n - dimensional tiles for any n ≥3. For each n , we show that a certain local move is ergodic, and conjecture that it has a mixing time of O ( L n +2 log L ) on regions of size L . For n =3, the tiles are rhombohedra, and the local move consists of switching between two tilings of a rhombic dodecahedron. We use coupling from the past to sample random tilings of a large rhombic dodecahedron, and show that arctic regions exist in which the tiling is frozen into a fixed state. However, unlike the two-dimensional tiling in which the arctic region is an inscribed circle, here it seems to be octahedral. In addition, height fluctuations between the boundary of the region and the center appear to be constant rather than growing logarithmically. We conjecture that this is because the physics of the model is in a ``smooth'' phase where it is rigid at large scales, rather than a ``rough'' phase in which it is elastic. Kasper K. Berthelsen and Jesper Møller. Spatial jump processes and perfect simulation, 2001. Preprint.
Abstract: Spatial birth-and-death processes, spatial birth-and-catastrophe processes, and more general types of spatial jump processes are studied in detail. Particularly, varied kind of coupling constructions are considered, leading to some known and some new perfect simulation procedures for different types of spatial jump processes in equilibrium. The considered equilibrium distributions include many classical Gibbs point process models and a new class of models for spatial point processes introduced in the text. Jesper Møller and Rasmus P. Waagepetersen. Simulation-based inference for spatial point processes, 2001. Preprint. Kasper K. Berthelsen and Jesper Møller. A primer on perfect simulation for spatial point processes, 2001. Preprint.
Abstract: This primer provides a self-contained exposition of the case where spatial birth-and-death processes are used for perfect simulation of locally stable point processes. Particularly, a simple dominating coupling from the past (CFTP) algorithm and the CFTP algorithms introduced in Kendall (1998), Kendall & Møller (2000), and Fernández, Ferrari & Garcia (2000) are studied. Some empirical results for the algorithms are discussed. Julian Besag. Likelihood analysis of binary data in space and time. Manuscript, 2001.
Keywords: Autologistic; Conditional probability; Coupling from the past; Lattice data; Markov chain Monte Carlo; Markov random fields; Maximum likelihood; Maximum pseudolikelihood; Perfect simulation; Space and time; Spread of disease David B. Wilson. Mixing times of lozenge tiling and card shuffling Markov chains, 2001. arXiv:math. PR/0102193.
Abstract: We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an L X L region we bound the mixing time by O ( L 4 log L ), which improves on the previous bound of O ( L 7 ), and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Saloff-Coste, by lower bounding the mixing time of various card-shuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an O ( n 3 log n ) upper bound on the mixing time of the Karzanov-Khachiyan Markov chain for linear extensions.
Remarks: Among other things this article analyses the running time of CFTP when applied to lozenge tilings, and contains experiments pertinent to Fill's algorithm. Kasper K. Berthelsen and Jesper Møller. Perfect simulation and inference for spatial point processes. Manuscript, 2002.
Abstract: The advantages and limitations of using perfect simulation for simulation-based inference for pairwise interaction point processes are discussed. Various aspects of likelihood inference for the Strauss process with unknown range of interaction are studied. A large part of the paper concerns non-parametric Bayesian inference for the interaction function. Markov chain Monte Carlo methods, particularly path sampling, play an important role. Several empirical results and various datasets are considered. Duncan J. Murdoch and Xiao-Li Meng. Towards perfect sampling for Bayesian mixture priors. Bayesian Methods with Applications to Science, Policy, and Official Statistics. Selected papers from ISBA 2000: the sixth world meeting of the International Society for Bayesian Analysis.
Abstract: Perfect sampling using the coupling from the past (CFTP) algorithm was introduced by Propp and Wilson in 1996. In much the way rejection sampling allows one to convert samplers from one distribution into samplers from another, CFTP allows one to convert Markov chain Monte Carlo algorithms from approximate samplers of the steady-state distribution into perfect ones. Since 1996 CFTP has been applied to many different Markov chains. However, its use in routine Bayesian computation is still in the early stages of development. This paper provides a couple of building blocks for its potentially routine application in Bayesian mixture priors, including a t mixture coupler, and demonstrates the types of difficulties that currently prevent CFTP from being applied routinely in Bayesian computation. Radu V. Craiu and Xiao-Li Meng. Antithetic coupling for perfect sampling. Bayesian Methods with Applications to Science, Policy, and Official Statistics. Selected papers from ISBA 2000: the sixth world meeting of the International Society for Bayesian Analysis.
Abstract: This paper reports some initial investigations of the use of antithetic variates in perfect sampling. A simple random walk example is presented to il-lustrate the key ingredients of antithetic coupling for perfect sampling as well as its potential benefit. A key step in implementing antithetic coupling is to generate k ≥2 random variates that are negatively associated, a stronger condition than negative correlation as it requires that the variates remain non-positively corre-lated after any (component-wise) monotone transformations have been applied. For k =2, this step is typically trivial (e. g., by taking U and 1- U where U.
U (0,1)) and it constitutes much of the common use of antithetic variates in Monte Carlo simulation. Our emphasis is on k >2 because we have observed some general gains in going beyond the commonly used pair of antithetic variates. We discuss several ways of generating negatively associated random variates for arbitrary k , and our comparison generally favors Iterative Latin Hypercube Sampling . Nancy L. Garcia. Perfect simulation of spatial processes. Resenhas - IME-USP, 4(3):281--324, 2000.
Abstract: This work presents a review of some of the schemes used to perfect sample from spatial processes. As examples, the area-interaction point process, Strauss process, penetrable sphere model, Peierls contours of the Ising model and continuous loss networks are studied under Coupling from The Past Algorithm (CFTP), Acceptance and Rejection Algorithm (ARA) and Backward-Forward Algorithm (BFA). Sung-woo Cho and Ashish Goel. Exact sampling in machine scheduling problems. Lecture Notes in Computer Science 2129 (Approximation Randomization and Combinatorial Optimization: Algorithms and Techniques), Michel X. Goemans, Klaus Jansen, José D. P. Rolim, Luca Trevisan, Eds., pages 202--210.
Abstract: Analysis of machine scheduling problem can get very complicated even for simple scheduling policies and simple arrival processes. The problem becomes even harder if the scheduler and the arrival process are complicated, or worse still, given to us as a black box. In such cases it is useful to obtain a typical state of the system which can then be used to deduce information about the performance of the system or to tune the parameters for either the scheduling rule or the arrival process. We consider two general scheduling problems and present an algorithm for extracting an exact sample from the stationary distribution of the system when the system forms an ergodic Markov chain. We assume no knowledge of the internals of the arrival process or the scheduler. Our algorithm assumes that the scheduler has a natural monotonic property, and that the job service times are geometric/exponential. We use the Coupling From The Past paradigm due to Propp and Wilson to obtain our result. In order to apply their general framework to our problems, we perform a careful coupling of the different states of the Markov chain. Ashish Goel and Michael Mitzenmacher. Exact sampling of TCP window states. To appear in IEEE Infocom, 2002.
Abstract: We demonstrate how to apply Coupling from the Past, a simulation technique for exact sampling, to Markov chains based on TCP variants. This approach provides a new, statistically sound paradigm for network simulations: instead of simulating a protocol over long times, or explicitly finding the stationary distribution of a Markov chain, use Coupling from the Past to quickly obtain samples from the stationary distribution. Coupling from the Past is most efficient when the underlying state space satisfies a partial order and certain monotonicity conditions. To efficiently apply this general paradigm to TCP, we demonstrate that the states of a simple TCP model possess a monotonic partial order; this order appears interesting in its own right. Preliminary simulation results indicate that this approach is quite efficient, and produces results which are similar to those obtained by simulating a TCP-Tahoe connection. Keith Crank and James A. Fill. Interruptible exact sampling in the passive case, 2002. arXiv:math. PR/0202136.
Abstract: We establish, for various scenarios, whether or not interruptible exact stationary sampling is possible when a finite-state Markov chain can only be viewed passively. In particular, we prove that such sampling is not possible using a single copy of the chain. Such sampling is possible when enough copies of the chain are available, and we provide an algorithm that terminates with probability one. Robert P. Dobrow and James A. Fill. Speeding up the FMMR perfect sampling algorithm: A case study revisited. Random Structures and Algorithms 23(4):434--452, 2003. arXiv:math. PR/0205120.
Abstract: In a previous paper by the second author, two Markov chain Monte Carlo perfect sampling algorithms - one called coupling from the past (CFTP) and the other (FMMR) based on rejection sampling - are compared using as a case study the move-to-front (MTF) self-organizing list chain. Here we revisit that case study and, in particular, exploit the dependence of FMMR on the user-chosen initial state. We give a stochastic monotonicity result for the running time of FMMR applied to MTF and thus identify the initial state that gives the stochastically smallest running time; by contrast, the initial state used in the previous study gives the stochastically largest running time. By changing from worst choice to best choice of initial state we achieve remarkable speedup of FMMR for MTF; for example, we reduce the running time (as measured in Markov chain steps) from exponential in the length n of the list nearly down to n when the items in the list are requested according to a geometric distribution. For this same example, the running time for CFTP grows exponentially in n . Anne Philippe and Christian P. Robert. Perfect simulation of positive Gaussian distributions. Statistics and Computing 13(2):179--186, 2003. (Postscript.)
Abstract: We provide an exact simulation algorithm that produces variables from truncated Gaussian distributions on R + p via a perfect sampling scheme, based on stochastic ordering and slice sampling, since accept-reject algorithms like the one of Geweke (1991) and Robert (1995) are difficult to extend to higher dimensions. Gareth O. Roberts and Jeffrey S. Rosenthal. Combinatorial identities associated with CFTP, 2002.
Abstract: We explore a method of obtaining combinatorial identities by analysing partially-completed runs of the Coupling from the Past (CFTP) algorithm. In particular, using CFTP for simple symmetric random walk (ssrw) with holding boundaries, we derive an identity involving linear combinations of C ab (s) for different a and b , where C ab (s) is the probability that unconstrained ssrw run from 0 for time n has maximum value a , and minimum value b , and ends up at s at time n . L. A. Breyer and G. O. Roberts. A new method for coupling random fields. LMS Journal of Computation and Mathematics 5:77--94, 2002.
Abstract: Given a Markov chain, a stochastic flow which simultaneously constructs sample paths started at each possible initial value can be constructed as a composition of random fields. In this paper, we describe a method for coupling flows by modifying an arbitrary field (consistent with the Markov chain of interest) by an independence Metropolis Hastings iteration. We show that the resulting stochastic flow has many desirable coalescence properties, regardless of the form of the original flow. L. A. Breyer and G. O. Roberts. Catalytic perfect simulation. Methodology and Computing in Applied Probability 3(2):161--177, 2001. Abstract: We introduce a methodology for perfect simulation using so called catalysts to modify random fields. Our methodology builds on a number of ideas previously introduced by Breyer and Roberts (1999), by Murdoch (1999), and by Wilson (2000). We illustrate our techniques by simulating two examples of Bayesian posterior distributions. David J. C. MacKay. Information Theory, Inference, and Learning Algorithms. Cambridge University Press (2003).
Textbook (viewable online), 640 pages. Chapters include.
29 Monte Carlo Methods.
30 Efficient Monte Carlo Methods.
31 Ising Models.
32 Exact Monte Carlo Sampling.
Chapter 32 is a 9-page introduction to exact sampling. Graeme K. Ambler. Dominated Coupling From The Past and Some Extensions of the Area-Interaction Process. PhD thesis, University of Bristol, Department of Mathematics, 2002.
Abstract: Dominated coupling from the past, an example of perfect simulation, is a method for sampling from the equilibrium distribution of a Markov chain. A particular model which may be sampled in this way is the area-interaction point process of Baddeley and van Lieshout (1995). This thesis reviews perfect simulation and some aspects of point process theory. It also reviews some methods for simulating spatial point processes. It then introduces an extension of the area-interaction process which incorporates both attractive and repulsive components at different scales. The properties of this model are investigated using some standard test functions, and the model is fitted to a standard data set. Some discrete analogues of these models are then discussed along with methods of sampling from these models using dominated coupling from the past. While discussing these sampling methods, we highlight some of the pitfalls that exist when one attempts to use dominated coupling from the past. We set out and investigate the use of the discrete analogue of the area-interaction process as a prior distribution on wavelet coefficients in Bayesian non-parametric regression. Finally, we discuss some of the issues that arose while implementing the algorithms, both for simulating from the attractive-repulsive model and for the posterior distribution of the Bayesian wavelet thresholding model. Jordi Artйs. Simulaciуn exacta en procesos puntuales espaciales: mйtodo CFTP / Exact simulation for spatial point processes: CFTP method. Master's thesis, Universitat Jaume I, Departamento de Matemбticas, 2004.
Abstract: We consider algorithms that sample exactly from probability distributions on continuous state spaces. For the most part we focus our attention on multivariate distributions. Examples and computer implementations are included for the algorithms considered. Algorithms discussed include Murdoch and Green's multigamma and rejection couplers and Fill and Huber's randomness recycler. We prove convergence in finite time for the multigamma and rejection couplers as described by Murdoch and Green, and adapt these to a multivariate context. Applications of the Fill/Huber randomness recycler to multivariate distributions on continuous state spaces are also considered. Other algorithms such as Wilson's multishift coupler are also discussed. Can Mert. Bayesian inference in Schizophrenia research. Master's thesis, Royal Institute of Technology, Computer Science, 2002.
Advisor: Stefan Arnborg.
Abstract: In this Master's thesis, Bayesian data mining techniques are used to analyze a database containing information from schizophrenia affected and healthy persons. The aim is to find dependencies between the diagnosis and various attributes. The database is assembled in the HUBIN project at the Karolinska Institutet. Two different techniques are employed, classification and graphical modeling. We also propose an MCMC extension to the graphical model problem. The results in this thesis indicate that the data mining approach is very promising in this kind of research. Several known facts about schizophrenia were reproducable in this study. Tina Dam, Jan Brønnum Sørensen, and Michael Hedegaard Thomsen. Master's thesis, Aalborg University, Department of Mathematical Sciences, 1999.
Abstract: This thesis presents the theory of spatial point processes. It introduces the general setting and characterization results for point processes, followed by definitions of the binomial and the Poisson point processes. The Poisson point process is important for the theory of point processes as it models complete spatial randomness. Various summary statistics are presented, and estimation and examples of the use of such are given. Emphasis is placed on Ripley's K-function and statistics related to it. Different classes of models are defined; Cox, cluster, hard-core, Markov, nearest neighbour Markov, and Gibbs processes and simulation of each of these models is explained. Markov chain Monte Carlo simulation is covered and here the Metropolis-Hastings algorithm is an important tool. We then present spatial birth.
Abstract: Perfect sampling algorithms are Markov Chain Monte Carlo (MCMC) methods without statistical error. The latter are used when one needs to get samples from certain (non-standard) distributions. This can be accomplished by creating a Markov chain that has the desired distribution as its stationary distribution, and by running sample paths "for a long time", i. e. until the chain is believed to be in equilibrium. The question "how long is long enough?" is generally hard to answer and the assessment of convergence is a major concern when applying MCMC schemes. This issue completely vanishes with the use of perfect sampling algorithms which -- if applicable -- enable exact simulation from the stationary distribution of a Markov chain. In this thesis, we give an introduction to the general idea of MCMC methods and perfect sampling. We develop advances in this area and highlight applications of these advances to two relevant problems. As advances, we discuss and devise several variants of the well-known Metropolis-Hastings algorithm which address accuracy, applicability, and computational cost of this method. We also describe and extend the idea of slice coupling, a technique which enables one to couple continuous sample paths of Markov chains for use in perfect samplingalgorithms. As a first application, we consider Bayesian variable selection. The problem of variable selection arises when one wants to model the relationship between a variable of interest and a subset of explanatory variables. In the Bayesian approach one needs to sample from the posterior distribution of the model and this simulation is usually carried out using regular MCMC methods. We significantly expand the use of perfect sampling algorithms within this problem using ideas developed in this thesis. As a second application, we depict the use of these methods for the interacting fermion problem. We employ perfect sampling for computing self energy through Feynman diagrams using Monte Carlo integration techniques. We conclude by stating two open questions for future research. Mariana Rodrigues Motta. Perfect Simulation of the Bi-dimentional Silo Model. Master's thesis, 2001.
Advisor: Nancy Lopes Garcia Murali Haran. Efficient Perfect and MCMC Sampling Methods for Bayesian Spatial and Components of Variance Models. PhD thesis, University of Minnesota, School of Statistics, 2003.
Author's description: I worked on Bayesian computation for spatial models and some hierarchical models. In particular, I investigated sampling techniques for hierarchical models that use Gaussian Markov Fields (GMRFs). A major portion of my thesis contains work on perfect sampling approaches for Bayesian disease mapping models. Graeme K. Ambler and Bernard W. Silverman. Perfect simulation for Bayesian wavelet thresholding with correlated coefficients, 2004. (PDF)
Abstract: We introduce a new method of Bayesian wavelet shrinkage for reconstructing a signal when we observe a noisy version. Rather than making the usual assumption that the wavelet coefficients of the signal are independent, we allow for the possibility that they are locally correlated in both location (time) and scale (frequency). This leads us to a prior structure which is, unfortunately, analytically intractable. Nevertheless, it is possible to draw independent samples from a close approximation to the posterior distribution by an approach based on Coupling From The Past, making it possible to use a simulation-based approach to fit the model. Graeme K. Ambler and Bernard W. Silverman. Perfect simulation of spatial point processes using dominated coupling from the past with application to a multiscale area-interaction point process, 2004. (PDF)
Abstract: We consider perfect simulation algorithms for locally stable point processes based on dominated coupling from the past. A new version of the algorithm is developed which is feasible for processes which are neither purely attractive nor purely repulsive. Such processes include multiscale area-interaction processes, which are capable of modelling point patterns whose clustering structure varies across scales. We prove correctness of the algorithm and existence of these processes. An application to the redwood seedlings data is discussed. M. N. M. van Lieshout, and R. S. Stoica. Perfect Simulation for Marked Point Processes. PNA-R0306, ISSN 1386-3711, 2003. (PDF)
Abstract: This paper extends some recently proposed exact simulation algorithms for point processes to marked patterns and reports on a simulation study into the relative efficiency for a range of Markov marked point processes. Elke Thönnes. Generic CFTP in Stochastic Geometry and beyond, Revista de Estatistica, Special Issue, Proceedings of the 23rd European Meeting of Statistician in Funchal, 81 - 84, 2001. Mark Huber. Perfect sampling using bounding chains. Annals of Applied Probability 14(2):734--753, 2004.
Abstract: Bounding chains are a technique that offers three benefits to Markov chain practitioners: a theoretical bound on the mixing time of the chain under restricted conditions, experimental bounds on the mixing time of the chain that are provably accurate and construction of perfect sampling algorithms when used in conjunction with protocols such as coupling from the past. Perfect sampling algorithms generate variates exactly from the target distribution without the need to know the mixing time of a Markov chain at all. We present here the basic theory and use of bounding chains for several chains from the literature, analyzing the running time when possible. We present bounding chains for the transposition chain on permutations, the hard core gas model, proper colorings of a graph, the antiferromagnetic Potts model and sink free orientations of a graph. Wilfrid S. Kendall. Notes on Perfect Simulation, 2004.
Abstract: Perfect simulation refers to the art of converting suitable Markov Chain Monte Carlo algorithms into algorithms which return exact draws from the target distribution, instead of long-time approximations. The theoretical concepts underlying perfect simulation have a long history, but they were first drawn together to form a practical simulation technique in the ground-breaking paper of Propp and Wilson [75], which showed (for example) how to obtain exact draws from (for example) the critical Ising model. These lecture notes are organized around four main themes of perfect simulation: the original or classic Coupling From The Past algorithm (CFTP); variations which exploit regeneration ideas such as small-set or split-chain constructions from Markov chain theory (small-set CFTP); generalizations of CFTP which deal with non-monotonic and non-uniformly ergodic examples (dominated CFTP); and finally some theoretical complements. Mark Huber. A bounding chain for Swendsen-Wang. Random Structures and Algorithms 22(1):43--59, 2002.
Abstract: The greatest drawback of Monte Carlo Markov chain methods is lack of knowledge of the mixing time of the chain. The use of bounding chains solves this difficulty for some chains by giving theoretical and experimental upper bounds on the mixing time. Moreover, when used with methodologies such as coupling from the past, bounding chains allow the user to take samples drawn exactly from the stationary distribution without knowledge of the mixing time. Here we present a bounding chain for the Swendsen-Wang process. The Swendsen-Wang bounding chain allow us to efficiently obtain exact samples from the ferromagnetic Q-state Potts model for certain classes of graphs. Also, by analyzing this bounding chain, we will show that Swendsen-Wang is rapidly mixing over a slightly larger range of parameters than was known previously. Nancy L. Garcia and Nevena Maric. Perfect simulation for a continuous one-dimensional loss network, 2002. (Postscript)
Abstract: Sufficient conditions for ergodicity of a one-dimensional loss networks on R with length distribution G and cable capacity C are found. These processes are spatial birth-and-death processes with an invariant measure which is absolutely continuous with respect to a Poisson process and we implement the perfect simulation scheme based on the clan of ancestors introduced by Fernández, Ferrari and Garcia (2002) to obtain perfect samples viewed in a finite window of the infinite-volume invariant measure. Moreover, by a better understanding of the simulation process it is possible to get a better condition for ergodicity. Yuzhi Cai and Wilfrid S. Kendall. Perfect simulation for correlated Poisson random variables conditioned to be positive. Statistics and Computing 12(3):229--243, 2002.
Abstract: In this paper we present a perfect simulation method for obtaining perfect samples from collections of correlated Poisson random variables conditioned to be positive. We show how to use this method to produce a perfect sample from a Boolean model conditioned to cover a set of points: in W. S. Kendall and E. Thцnnes (Pattern Recognition 32(9):1569--1586, 1999), this special case was treated in a more complicated way. The method is applied to several simple examples where exact calculations can be made, so as to check correctness of the program using χ²-tests, and some small-scale experiments are carried out to explore the behaviour of the conditioned Boolean model. Ian W. McKeague and Marc Loizeaux. Perfect Sampling for Point Process Cluster Modelling. Chapter 5 of Spatial Cluster Modelling (A. Lawson and D. Denison, eds.), Chapman and Hall (2002), pages 87-107. (Postscript) Marc A. Loizeaux and Ian W. McKeague. Perfect Sampling for Posterior Landmark Distributions with an Application to the Detection of Disease Clusters. IMS Lecture Notes - Monograph Series, Vol. 37, 321-331 (2001). (Postscript)
Abstract: We study perfect sampling for the posterior distribution in a class of spatial point process models introduced by Baddeley and van Lieshout (1993). For Neyman--Scott cluster models, perfect sampling from the posterior is shown to be computationally feasible via a coupling-from-the-past type algorithm of Kendall and Møller. An application to data on leukemia incidence in upstate New York is presented. M. N. M. van Lieshout and Erik van Zwet. Exact sampling from conditional Boolean models with applications to maximum likelihood inference. Advances in Applied Probabability 33(2):339--353, 2001.
Abstract: We are interested in estimating the intensity parameter of a Boolean model of discs (the bombing model) from a single realization. To do so, we derive the conditional distribution of the points (germs) of the underlying Poisson process. We demonstrate how to apply coupling from the past to generate samples from this distribution, and use the samples thus obtained to approximate the maximum likelihood estimator of the intensity. We discuss and compare two methods: one based on a Monte Carlo approximation of the likelihood function, the other a stochastic version of the EM algorithm. George Casella, Christian P. Robert, and Martin T. Wells. Rao-Blackwellization of Generalized Accept-Reject Schemes, 2000. (Postscript)
Abstract: This paper extends the accept-reject algorithm to allow the proposal distribution to change at each iteration. We first establish a necessary and sufficient condition for this generalized accept-reject algorithm to be valid, and then show how the Rao-Blackwellization of Casella and Robert (1996) can be extended to this setting. An important application of these results is to the perfect sampling technique of Fill (1998), which is a generalized accept-reject algorithm in disguise. Jesper Møller and Jakob G. Rasmussen. Perfect Simulation of Hawkes Processes. Research Report R-2004-18, Department of Mathematical Sciences, Aalborg University, 2004.
Abstract: This article concerns a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structure, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. Examples of applications and empirical results are presented. Wilfrid S. Kendall. Geometric Ergodicity and Perfect Simulation, 2004. (PDF)
Abstract: This note extends the work of Foss and Tweedie (1998), who showed that availability of the classic Coupling from The Past algorithm of Propp and Wilson (1996) is essentially equivalent to uniform ergodicity for a Markov chain (see also Hobert and Robert 2004). In this note we show that all geometrically ergodic chains possess dominated Coupling from The Past algorithms (not necessarily practical!) which are rather closely connected to Foster-Lyapunov criteria. Chris Holmes and David G. T. Denison. Pefect sampling for wavelet reconstruction of signals. IEEE Transactions on Signal Processing 50(2):337--344, 2002. (PDF)
Abstract: The coupling from the past (CFTP) procedure is a protocol for finite-state Markov chain Monte Carlo (MCMC) methods whereby the algorithm itself can determine the necessary runtime to convergence. In this paper, we demonstrate how this protocol can be applied to the problem of signal reconstruction using Bayesian wavelet analysis where the dimensionality of the wavelet basis set is unknown, and the observations are distorted by Gaussian white noise of unknown variance. MCMC simulation is used to account for model uncertainty by drawing samples of wavelet bases for approximating integrals (or summations) on the model space that are either too complex or too computationally demanding to perform analytically. We extend the CFTP protocol by making use of the central limit theorem to show how the algorithm can also monitor its own approximation error induced by MCMC. In this way, we can assess the number of MCMC samples needed to approximate the integral to within a user specified tolerance level. Hence, the method automatically ensures convergence and determines the necessary number of iterations needed to meet the error criteria. Radu V. Craiu and Xiao-Li Meng. Multi-process parallel antithetic coupling for backward and forward Markov chain Monte Carlo. Preprint, 2003.
Abstract: Antithetic coupling is a general stratification strategy for reducing Monte Carlo variance without increasing the simulation size. The use of the antithetic principle in the Monte Carlo literature typically employs two strata via antithetic quantile coupling. We demonstrate here that further stratification, obtained by using k > 2 (e. g., k = 3--10) antithetically coupled variates, can offer substantial additional gain in Monte Carlo efficiency, in terms of both variance and bias. The reason for reduced bias is that antithetically coupled chains can provide a more dispersed search of the state space than multiple independent chains. The emerging area of perfect simulation provides a perfect setting for implementing the k-process parallel antithetic coupling for MCMC because, without antithetic coupling, this class of methods delivers genuine independent draws. Furthermore, antithetic backward coupling provides a very convenient theoretical tool for investigating antithetic forward coupling. However, the generation of k > 2 antithetic variates that are negatively associated , i. e., they preserve negative correlation under monotone transformations, and extremely antithetic , i. e., they are as negatively correlated as possible, is more complicated compared to the case with k = 2. In this paper, we establish a theoretical framework for investigating such issues. Among the generating methods that we compare, Latin hypercube sampling and its iterative extension appear to be general-purpose choices, making another direct link between Monte Carlo and Quasi Monte Carlo. J. N. Corcoran and U. Schneider. Pseudo-perfect and adaptive variants of the Metropolis-Hasting algorithm with an independent candidate density, 2004. To appear in Journal of Statistical Computation and Simulation . (PDF)
Abstract: We describe and examine an imperfect variant of a perfect sampling algorithm based on the Metropolis-Hastings algorithm that appears to perform better than a more traditional approach in terms of speed and accuracy. We then describe and examine an ``adaptive'' Metropolis-Hasting algorithm which generates and updates a self-target candidate density in such a way that there is no ``wrong choice'' for an initial candidate density. Simulation examples are provided. M. S. Carvalho and J. N. Corcoran. Solving stochastic difference equations originated from continuous-time threshold AR(1) models through perfect simulation, 2003. Preprint.
Abstract: In this paper we look at a discretization of a continuous-time threshold AR(1) process (CTAR(1)) and show how to apply perfect simulation to find the stationary distribution of the resulting stochastic difference equation. Such method leads to perfect solutions of stochastic difference equations of this kind. Depending on how sensitive the original CTAR(1) process is to the discretization, the method may give an approximate solution to the original continuous-time process. Also, we introduce the phenomenon of decoupling in a perfect simulation algorithm. Jesper Møller, A. N. Pettitt, K. K. Berthelsen, and R. W. Reeves. An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. (PDF)
Abstract: We present new methodology for drawing samples from a posterior distribution when (i) the likelihood function or (ii) a part of the prior distribution is only specified up to a normalising constant. In the case (i), the novelty lies in the introduction of an auxiliary variable in a Metropolis-Hastings algorithm and the choice of proposal distribution so that the algorithm does not depend upon the unknown normalising constant. In the case (ii), similar ideas apply and the situation is even simpler as no auxiliary variable is required. Our method is ``on-line'' as compared with alternative approaches to the problem which require ``off-line'' computations. Since it is needed to simulate from the ``unknown distribution'', e. g. the likelihood function in case (i), perfect simulation such as the Propp-Wilson algorithm becomes useful. We illustrate the method in case (i) by producing posterior samples when the likelihood is given by an Ising model and by a Strauss point process. Shane G. Henderson and Richard L. Tweedie. Perfect simulation for Markov chains arising from discrete-event simulation. Proceedings of the 38th Annual Allerton Conference on Communication, Control and Computing, pp. 1125--1134, 2000. (Postscipt)
Abstract: Virtually any discrete-event simulation can be rigorously defined as a Markov chain evolving on a general state space, and under appropriate conditions, the chain has a unique stationary probability distribution. Many steady-state performance measures can be expressed in terms of the stationary probability distribution of the chain. We would like to apply ``coupling from the past'' algorithms to obtain samples from the stationary probability distribution of such chains. Unfortunately, the structural properties of the chains arising from discrete-event simulations preclude the immediate application of current coupling from the past algorithms. We describe why this is the case, and extend a class of coupling from the past algorithms so that they may be applied in this setting. Michael Korn and Igor Pak. Tilings of rectangles with T-tetrominoes, 2003. Theoretical Computer Science 319(1-3):3-27, 2004. [PDF]
Abstract: We prove that any two tilings of a rectangular region by T-tetrominoes are connected by moves involving only 2 and 4 tiles. We also show that the number of such tilings is an evaluation of the Tutte polynomial. The results are extended to a more general class of regions.
From the introduction: Along the way we show that our new height functions enable us to define a lattice structure on all T - tetromino tilings of a rectangle. While this may seem a theoretical curiosity, in fact there are important applications of this result. There is a natural definition of a Markov chain on tilings (perform ``random local moves'') which translates into a nice Markov chain on height functions. Using the ``coupling from the past'' technique of Propp and Wilson and the fact that the height functions form a lattice (see [15, 14]) one can sample random T-tetromino tilings from an exactly uniform distribution (as opposed to a ``nearly uniform'' distribution usually obtained by the Markov chain approach). We refer to [16] for a full discussion of this approach and other examples of lattice structures on tilings. James P. Hobert and Christian P. Robert. A mixture representation of π with applications in Markov chain Monte Carlo and perfect sampling. The Annals of Applied Probability 14(3):1295--1305, 2004. (PDF)
Abstract: Let X = n : n =0 1 2 . > be an irreducible, positive recurrent Markov chain with invariant probability measure π. We show that if X satisfies a one-step minorization condition, then π can be represented as an infinite mixture. The distributions in the mixture are associated with the hitting times on an accessible atom introduced via the splitting construction of Athreya and Ney [ Trans. Amer. Matemática. Soc. 245 (1978) 493--501] and Nummelin [ Z. Wahrsch. Verw. Gebiete 43 (1978) 309--318]. When the small set in the minorization condition is the entire state space, our mixture representation of π reduces to a simple formula, first derived by Breyer and Roberts [ Methodol. Comput. Appl. Probab. 3 (2001) 161--177] from which samples can be easily drawn. Despite the fact that the derivation of this formula involves no coupling or backward simulation arguments, the formula can be used to reconstruct perfect sampling algorithms based on coupling from the past (CFTP) such as Murdoch and Green's [ Scand. J. Statist. 25 (1998) 483--502] Multigamma Coupler and Wilson's [ Random Structures Algorithms 16 (2000) 85--113] Read-Once CFTP algorithm. In the general case where the state space is not necessarily 1-small, under the assumption that X satisfies a geometric drift condition, our mixture representation can be used to construct an arbitrarily accurate approximation to π from which it is straightforward to sample. One potential application of this approximation is as a starting distribution for a Markov chain Monte Carlo algorithm based on X . Stephen P. Brooksy, Yanan Fan, and Jeffrey S. Rosenthal. Perfect Forward Simulation via Simulated Tempering. Pré-impressão.
Abstract: Several authors discuss how the simulated tempering scheme provides a very simple mechanism for introducing regenerations within a Markov chain. In this paper we explain how regenerative simulated tempering schemes provide a very natural mechanism for perfect simulation. We use this to provide a perfect simulation algorithm, which uses a single-sweep forward-simulation without the need for recursively searching through negative times. We demonstrate this algorithm in the context of several examples. F. Amaya and J. M. Benedí. 2000. Using Perfect Sampling in Parameter Estimation of a Wole Sentence Maximum Entropy Language Model. Proc. Fourth Computational Natural Language Learning Workshop, CoNLL-2000. Pages 79-82.
Abstract: The Maximum Entropy principle (ME) is an appropriate framework for combining information of a diverse nature from several sources into the same language model. In order to incorporate long-distance information into the ME framework in a language model, a Whole Sentence Maximum Entropy Language Model (WSME) could be used. Until now MonteCarlo Markov Chains (MCMC) sampling techniques has been used to estimate the paramenters of the WSME model. In this paper, we propose the application of another sampling technique: the Perfect Sampling (PS). The experiment has shown a reduction of 30% in the perplexity of the WSME model over the trigram model and a reduction of 2% over the WSME model trained with MCMC. Shuji Kijima and Tomomi Matsui. Polynomial time perfect sampling algorithm for two-rowed contingency tables, 2003. Tech report METR 2003-15.
Abstract: This paper proposes a polynomial time perfect (exact) sampling algorithm for 2×n contingency tables. Our algorithm is a Las Vegas type randomized algorithm and the expected running time is bounded by O(n 3 ln N) where n is the number of columns and N is the total sum of whole entries in a table. The algorithm is based on monotone coupling from the past (monotone CFTP) algorithm and new Markov chain for sampling two-rowed contingency tables uniformly. We employed the path coupling method and showed that the mixing rate of our chain is bounded by n(n-1)²(1+ln(n N)). Our result indicates that uniform generation of two-rowed contingency tables is easier than the corresponding counting problem, since the counting problem is known to be #P-complete. Tomomi Matsui and Shuji Kijima. Polynomial Time Perfect Sampler for Discretized Dirichlet Distribution, 2003. Tech report METR 2003-17.
Abstract: In this paper, we propose a perfect (exact) sampling algorithm according to a discretized Dirichlet distribution. Our algorithm is a monotone coupling from the past algorithm, which is a Las Vegas type randomized algorithm. We propose a new Markov chain whose limit distribution is a discretized Dirichlet distribution. Our algorithm simulates transitions of the chain O(n 3 ln Δ) times where n is the dimension (the number of parameters) and 1/Δ is the grid size for discretization. Thus the obtained bound does not depend on the magnitudes of parameters. In each transition, we need to sample a random variable according to a discretized beta distribution (2-dimensional Dirichlet distribution). To show the polynomiality, we employ the path coupling method carefully and show that our chain is rapidly mixing. Motoya Machida. Fill's algorithm for absolutely continuous stochastically monotone kernels. Electronic Communications in Probability, Vol. 7, paper 15, 2002.
Abstract: Fill, Machida, Murdoch, and Rosenthal (2000) presented their algorithm and its variants to extend the perfect sampling algorithm of Fill (1998) to chains on continuous state spaces. We consider their algorithm for absolutely continuous stochastically monotone kernels, and show the correctness of the algorithm under a set of certain regularity conditions. These conditions succeed in relaxing the previously known hypotheses sufficient for their algorithm to apply. Nancy L. Garcia and Mariana R. Motta. Perfect simulation for a stationary silo with absorbing walls. Methodology and Computing in Applied Probability 5(1):109--121, 2003. (Postscript)
Abstract: The objective of this work is to generate random samples of the unique stationary distribution associated to the stochastic model for grain storage in a finite bidimensional silo. The support of this measure is an unbounded and continuous state space and therefore a truncation was necessary to apply the CFTP perfect simulation scheme. The performance of the algorithm was measure by comparing the sample moments to the theoretical ones. Wolfgang Hörmann and Josef Leydold. Improved perfect slice sampling, 2003. Preprint.
Abstract: Perfect slice sampling is a method to turn Markov Chain Monte Carlo (MCMC) samplers into exact generators for independent random variates. The originally proposed method is rather slow and thus several improvements have been suggested. However, two of them are erroneous. In this article we give a short introduction to perfect slice sampling, point out incorrect methods, and give a new improved version of the original algorithm. Damian Clancy and Philip D. O'Neill Perfect simulation for stochastic models of epidemics among a community of households, 2002. Preprint.
Abstract: Recent progress in the analysis of infectious disease data using stochastic epidemic models has been largely due to Markov chain Monte Carlo methodology. In the present paper it is illustrated that certain methods of perfect simulation can also be applied in the context of epidemic models. Specifically, a perfect simulation algorithm due to Corcoran and Tweedie (2002) is considered for epidemic models that feature a population divided into households. The models allow individuals to be potentially infected both from outside the household, and from other household members. The data are assumed to consist of the final numbers ultimately infected within each household in the community. The methods are applied to data taken from outbreaks of in and the results compared with those obtained by other approaches. R. Fernández, P. A. Ferrari, and A. Galves. Coupling, renewal and perfect simulation of chains of infinite order. Notes for a course in the V Brazilian School of Probability, Ubatuba 2001.
This is a 95 pages booklet developing an explicit construction of chains with infinite memory under the Harris regime. S. Foss and T. Konstantopoulos. Extended renovation theory and limit theorems for stochastic ordered graphs. Markov Processes Relat. Fields 9:413--468, 2003. (PDF)
Abstract: We extend Borovkov's renovation theory to obtain criteria for coup-ling - convergence of stochastic processes that do not necessarily obey stochastic recursions. The results are applied to an ``infinite bin model'', a particular system that is an abstraction of a stochastic ordered graph, i. e., a graph on the integers that has (i, j), i Francesco Bartolucci and Julian Besag. A recursive algorithm for Markov random fields. Biometrika 89(3):724--730, 2002. (PDF)
Abstract: We propose a recursive algorithm as a more useful alternative to the Brook expansion for the joint distribution of a vector of random variables when the original formulation is in terms of the corresponding full conditional distributions, as occurs for Markov random fields. Usually, in practical applications, the computational load will still be excessive but then the algorithm can be used to obtain the componentwise full conditionals of a system after marginalizing over some variables or the joint distribution of subsets of the variables, conditioned on values of the remainder, which is required for block Gibbs sampling. As an illustrative example, we apply the algorithm in the simplest nontrivial setting of hidden Markov chains. More important, we demonstrate how it applies to Markov random fields on regular lattices and to perfect block Gibbs sampling for binary systems. Olle Häggström, Johan Jonasson, and Russell Lyons. Coupling and Bernoullicity in random-cluster and Potts models. Bernoulli 8(3):275--294, 2002. (Postscript)
Abstract: An explicit coupling construction of random-cluster measures is presented. As one of the applications of the construction, the Potts model on amenable Cayley graphs is shown to exhibit at every temperature the mixing property known as Bernoullicity. M. N. M. van Lieshout, and A. J. Baddeley. Extrapolating and interpolating spatial patterns. Chapter 4 of Spatial Cluster Modelling (A. Lawson and D. Denison, eds.), Chapman and Hall/CRC, 2002. Pages 61--86.
Abstract: We discuss issues arising when a spatial pattern is observed within some bounded region of space, and one wishes to predict the process outside of this region (extrapolation) as well as to perform inference on features of the pattern that cannot be observed (interpolation). We focus on spatial cluster analysis. Here the interpolation arises from the fact that the centres of clustering are not observed. We take a Bayesian approach with a repulsive Markov prior, derive the posterior distribution of the complete data, i. e. cluster centres with associated o#spring marks, and propose an adaptive coupling from the past algorithm to sample from this posterior. The approach is illustrated by means of the redwood data set (Ripley, 1977). Jorge Mateu, Jordi Artés, and José A. López. Computational issues for perfect simulation in spatial point patterns. Communications in Nonlinear Science and Numerical Simulation 9(2):229--240, 2004. Petar M. Djurić, Yufei Huang, and Tadesse Ghirmai. Perfect sampling: a review and applications to signal processing. IEEE Transactions on Signal Processing 50(2):345--356, 2002. (PDF)
Abstract: In recent years, Markov chain Monte Carlo (MCMC) sampling methods have gained much popularity among researchers in signal processing. The Gibbs and the Metropolis-Hastings algorithms, which are the two most popular MCMC methods, have already been employed in resolving a wide variety of signal processing problems. A drawback of these algorithms is that in general, they cannot guarantee that the samples are drawn exactly from a target distribution. More recently, new Markov chain-based methods have been proposed, and they produce samples that are guaranteed to come from the desired distribution. They are referred to as perfect samplers. In this paper, we review some of them, with the emphasis being given to the algorithm coupling from the past (CFTP). We also provide two signal processing examples where we apply perfect sampling. In the first, we use perfect sampling for restoration of binary images and, in the second, for multiuser detection of CDMA signals. Tournois Florent. Perfect simulation of a stochastic model for CDMA coverage. INRIA research report #4348, 2002.
Abstract: The aim of this article is to present ways of getting realizations of the stochastic model for CDMA coverage introduced in [1]. This is a basic stochastic geometry model that represents the location of antennas as realizations of Poisson point processes in the plane and allows one to capture regions (CDMA cells) of the plane with signal-to-noise ratio SNR to/from a given antenna at a sufficient level. We describe an algorithm of perfect simulation of arbitrary close approximation of the the CDMA cell mosaic given some coverage constraints. This is an adaptation of the so called backward simulation of conditional coverage process developed in [4]. The approximation comes from the estimation of the SNR, that is modeled by Poisson shot-noise process, based on an influence window whose size is theoretically adjusted to meet the true value of SNR with an arbitrary precision. As a motivating example we estimate the distribution functions of the distance from a fixed location to regions under various given handoff conditions (so called contact distribution functions). Philip D. O'Neill. Perfect simulation for Reed-Frost epidemic models. Statistics and Computing 13(1):37--44, 2003.
Abstract: The Reed-Frost epidemic model is a simple stochastic process with parameter q that describes the spread of an infectious disease among a closed population. Given data on the final outcome of an epidemic, it is possible to perform Bayesian inference for q using a simple Gibbs sampler algorithm. In this paper it is illustrated that by choosing latent variables appropriately, certain monotonicity properties hold which facilitate the use of a perfect simulation algorithm. The methods are applied to real data. J. N. Corcoran and U. Schneider. Shift and scale coupling methods for perfect simulation. Probability in the Engineering and Informational Sciences 17(3):277--303, 2003. (PDF)
Abstract: We describe and develop a variation on a layered multishift coupler due to Wilson that uses a slice sampling procedure to allow one to obtain potentially common draws from two different distributions. Our main application is coupling sample paths of Markov chains for use in perfect sampling algorithms. The coupler is based on slicing density functions and we describe a ``folding'' mechanism as an attractive alternative to the accept/reject step commonly used in slice sampling algorithms. Applications of the coupler are given to storage models and to auto-gamma distribution sampling. U. Schneider and J. N. Corcoran. Perfect sampling for Bayesian variable selection in a linear regression model. Journal of Statistical Planning and Inference 126(1):153--171, 2004.
Abstract: We describe the use of perfect sampling algorithms for Bayesian variable selection in a linear regression model. Starting with a basic case solved by Huang and Djuric (2002), where the model coefficients and noise variance are assumed to be known, we generalize the model step by step to allow for other sources of randomness, specifying perfect simulation algorithms that solve these cases by incorporating various techniques including Gibbs sampling, the perfect independent Metropolis-Hastings (IMH) algorithm, and recently developed ``slice coupling'' algorithms. Applications to simulated data sets suggest that our algorithms perform well in identifying relevant predictor variables. J. N. Corcoran, U. Schneider, and H.-B. Schьttler. Perfect stochastic summation of high order Feynman graph expansions, 2003. Preprint.
Abstract: We describe a new application of an existing perfect sampling technique of Corcoran and Tweedie to estimate the self energy of an interacting Fermion model via Monte Carlo summation. Simulations suggest that the algorithm in this context converges extremely rapidly and results compare favorably to true values obtained by brute force computations for low dimensional toy problems. A variant of the perfect sampling scheme which improves the accuracy of the Monte Carlo sum for small samples is also given. Iain Murray, Zoubin Ghahramani, and David J. C. MacKay. MCMC for doubly-intractable distributions. Proceedings of the 22nd Annual Conference on Uncertainty in Artificial Intelligence (UAI) , 2006. (PDF)
Abstract: Markov Chain Monte Carlo (MCMC) algorithms are routinely used to draw samples from distributions with intractable normalization constants. However, standard MCMC algorithms do not apply to doubly-intractable distributions in which there are additional parameter-dependent normalization terms; for example, the posterior over parameters of an undirected graphical model. An ingenious auxiliary-variable scheme (Møller et al., 2004) offers a solution: exact sampling (Propp and Wilson, 1996) is used to sample from a Metropolis-Hastings proposal for which the acceptance probability is tractable. Unfortunately the acceptance probability of these expensive updates can be low. This paper provides a generalization of Møller et al. (2004) and a new MCMC algorithm, which obtains better acceptance probabilities for the same amount of exact sampling, and removes the need to estimate model parameters before sampling begins. James A. Fill and Mark Huber. Perfect simulation of Vervaat perpetuities. arXiv:0908.1733.
Abstract: We use coupling into and from the past to sample perfectly in a simple and provably fast fashion from the Vervaat family of perpetuities. The family includes the Dickman distribution, which arises both in number theory and in the analysis of the Quickselect algorithm, which was the motivation for our work.
Theses Relevant to Perfect Sampling.
A number of people have written graduate theses in which a significant component was on perfect sampling with Markov chains. In chronological order these are:
Tine Møller Sørensen. A study of Propp & Wilson's method for exact Markov chain Monte Carlo sampling, 1997. Master's thesis in computational statistics, University of Bath.
Morten Fismen. Exact simulation using Markov chains. Technical Report 6/98, Institutt for Matematiske Fag, 1998. Diploma-thesis.
Karin Nelander. Contributions to Percolation Theory and Exact Sampling. Ph. D. thesis, Deperartment of Mathematics, Göteborg, March 29, 1998.
Jem Noelle Corcoran. Perfect Sampling in MCMC Algorithms. Ph. D. thesis, Department of Statistics at Colorado State University, May 1998.
Bas Straatman. Exact sampling and applications to a mite dispersal model. Master's thesis, Utrecht, August 10, 1998.
Elke Thönnes. Perfect and Imperfect Simulations in Stochastic Geometry. Ph. D. thesis, University of Warwick, October 1998.
Mark Lawrence Huber. Perfect Sampling Using Bounding Chains. Ph. D. thesis, Cornell University, May 1999.
Advisor: Mats Rudemo.
Alison Gibbs. Convergence of Markov Chain Monte Carlo Algorithms with Applications to Image Restoration. PhD thesis, University of Toronto, Department of Statistics, October 1999. (Postscript, PDF.)
Jesper Møller. Aspects of spatial statistics, stochastic geometry and Markov chain Monte Carlo methods. Thesis for doctoral degree in Natural Sciences, Aalborg University.
Mariana Rodrigues Motta. Perfect Simulation of the Bi-dimentional Silo Model. Master's thesis, 2001.
Giovanni Montana. Small sets and domination in perfect simulation. Ph. D. thesis, University of Warwick, April 2002.
Graeme K. Ambler. Dominated Coupling From The Past and Some Extensions of the Area-Interaction Process. PhD thesis, University of Bristol, Department of Mathematics, 2002.
Advisor: Stefan Arnborg.
Murali Haran. Efficient Perfect and MCMC Sampling Methods for Bayesian Spatial and Components of Variance Models. PhD thesis, University of Minnesota, School of Statistics, 2003.
Jordi Artйs. Simulaciуn exacta en procesos puntuales espaciales: mйtodo CFTP / Exact simulation for spatial point processes: CFTP method. Master's thesis, Universitat Jaume I, Departamento de Matemбticas, 2004.
Tuomas Rajala. Traditional MCMC and Exact simulation of the Ising model. Master's thesis, University of Jyväskylä, Department of Mathematics and Statistics, 2005.
Advisors: Stefan Geiss and Antti Penttinen.
Research Talks.
There are a number of research talks that do not appear to have associated written articles that are publicly available. These are listed below.
Chris Jennison. Approximate exact sampling: towards the general application of Propp and Wilson's algorithm. Seminar at University of Bath, 1998.
Abstract: Propp and Wilson's sampling-from-the-past rule allows the possibility of exact sampling by Markov chain Monte Carlo (MCMC) methods. We describe the Propp and Wilson method and consider details of its practical usage.
We then consider a scheme for applying the basic idea of Propp and Wilson in general problems. The method becomes approximate once more but with arguable advantages over other MCMC samplers. Xiao-Li Meng. Improving Perfect Simulation. Seminar at the Department of Statistics, The University of Chicago, January 25, 1999.
The Monte Carlo community recently has had a pleasant surprise. Based on their ingenious idea of running coupled Markov chains with negative time, Propp and Wilson (1996, Random Structures and Algorithms ) proposed a Markov chain Monte Carlo (MCMC) algorithm that will, with probability one, terminate itself in a finite number of steps and yet deliver draws that are exactly from the limiting (i. e., stationary) distribution. By applying the algorithm to a variety of problems, including sampling exactly from the Ising model at the critical temperature with a 4200 by 4200 toroidal grid, they demonstrated the practicality and efficiency of the algorithm. Their findings are perhaps particularly exciting for statisticians/probabilists/applied mathematicians, since this appears to be the first time that a major Monte Carlo method is developed solely by non-physicists, yet the method is applicable to non-trivial computational problems in physics and other scientific studies. Built upon their method of coupling from the past or backward coupling , a new class of MCMC algorithms has emerged. This class of algorithms has been labeled perfect simulation or exact simulation , because for this class of stochastic iterative algorithms the challenging issue of accessing convergence as well as errors in approximation completely vanishes.
In this talk I will first give a brief tutorial of the Propp-Wilson algorithm and Murdoch and Green's (1998, Scandinavian Journal of Statistics ) extension to continuous-state Markov chains. I will then propose two methods, multi-stage backward coupling and parallel antithetic coupling , for improving the computational or equivalently statistical efficiency of the current perfect simulation schemes. The first method borrows ideas from survey sampling where it is a common knowledge that multi-stage sampling is typically more cost effective than single-stage sampling. The second method explores the use of antithetic variates with backward coupling, and it relies critically on the theory of negative association , which studies the preservation of negative correlation under monotone transformations. Limited empirical and theoretical evidence suggests the possibility of significant improvement (e. g., 30%-50% reduction in time/variance) with either method over the Propp-Wilson algorithm in certain applications.
While the theory and implementation of the proposed methods is not completely trivial, the talk should be accessible to both the pessimists (nothing is perfect) and the optimists (things can always be better).
Motoya Machida. The Use of Perfect Sampling for Mixture Problems. Talk given at the Fourth North American New Researchers Conference, held at The Johns Hopkins University, August 4 - 7, 1999, also given in Singapore.
Since the seminal work of Propp and Wilson on perfect sampling, two algorithms for perfect sampling are known, one based on coupling from the past (CFTP) by Propp and Wilson, and the other via rejection sampling by Fill. The idea of perfect sampling is, in essence, that given an ergodic Markov chain we can generate a perfect stationary observation within a finite number of steps. The CFTP algorithm has been studied and used in various ways along with applications in Markov Chain Monte Carlo (MCMC). However, the reports on similar investigations for Fill's algorithm are limited compared with the CFTP algorithm.
Recently Hobert, Robert, and Titterington studied the implementation of the CFTP algorithm for two - and three-component mixtures with known components and reported the performances of the CFTP algorithm for these cases. They showed that the MCMC for the two-component mixture is monotone with a natural linear ordering over the weights. By extending such coupling to the case of three-component mixture, they observed that the coupled chains are contained in a lozenge-shaped region, which simplifies the algorithm to detect coalescence. But their coupling method for $k$-component mixtures with $n$ samples requires $ >$ transitions at the initial step, no matter how much the size of transitions can be reduced later. Therefore, as they argued, it is difficult to extend their method further because of the combinatorial explosion.
The coalescence of all the coupled chains is the essential characteristic for the CFTP algorithm. On the other hand, Fill's algorithm for stochastically monotone case, in theory, does not require such grand coalescence. In fact, as shown in Fill's paper, the proof of his algorithm's validity relies only on the property of a bivariate coupling. In this talk, we exploit this property further and demonstrate how we can use it for mixture problems of more than three components.
Abstract: An MCMC sampling procedure known as coupling from the past (CFTP) is described for variable selection in the Bayesian linear model with orthogonal predictors, which includes wavelets, Fourier series and Chebyshev polynomials. CFTP allows one to draw perfect i. i.d. samples from the posterior model space in realistic CPU time. Examples are used to demonstrate this approach on a number of curve fitting problems. Olivier Roques. Coupler depuis le passй (Coupling from the past). Seminar, May 19, 1999.
Abstract: Lorsque l'on souhaite engendrer des objets selon une distribution P, on peut construire une chaine de Markov ergodique dont la distribution stationnaire est P. Tout le probleme est de savoir combien de temps il faut faire tourner ce processus pour approcher а epsilon pres la distribution stationnaire, i. e borner le temps de mйlange de la chaine. La mйthode "Couplage depuis le passй" [Kendall, Propp, Wilson, Jerrum. ] permet de s'abstenir de ces calculs souvent compliquйs et d'obtenir un йchantillon de l'ensemble des йtats, distibuй exactement selon la distribution stationnaire. Duncan Murdoch. Perfect Sampling: Not Just for Markov Chains? Talk 263-10 at the 1999 Joint Satistical Meeting, also given at McMaster.
Abstract: Propp and Wilson's (1996) coupling from the past (CFTP) algorithm generates a sample from the limiting distribution of a Markov chain. In this talk I argue that the underlying idea of CFTP is more widely applicable, and will demonstrate successful and unsuccessful attempts to apply it to problems ranging from approximation of integrals to simulation of stochastic differential equations. Antonietta Mira. Ordering Slicing and Splitting Monte Carlo Markov Chains. Talk 263-28 at the 1999 Joint Satistical Meeting.
Abstract: The class of Markov chains having a specified stationary distribution is very large, so it is important to have criteria telling when one chain performs better than another. The Peskun ordering is a partial ordering on this class. We study the implications of the Peskun ordering both in finite and general state spaces. Unfortunately there are many Metropolis-Hastings samplers that are not comparable in the Peskun sense. We thus define a more useful ordering, the covariance ordering, that allows us to compare a wider class of sampling schemes. It is always possible to beat the Metropolis-Hastings algorithm in terms of the Peskun ordering. Two new algorithms will be introduced and studied for this purpose. The slice sampler outperforms the independence Metropolis-Hastings algorithm. The splitting rejection sampler outperforms the more general Metropolis-Hastings algorithm. Finally we propose a ``perfect'' version of the slice sampler. Perfect sampling allows exact simulation of random variables from the stationary measure of a Markov chain. By exploiting monotonicity properties of the slice sampler we show that a perfect version of the algorithm can be easily implemented. Motoya Machida. Perfect Sampling via Cross-Monotonicity for Simple Mixture Problems. October 13, 1999.
Abstract: Since Propp and Wilson's seminal work published in 1996, the perfect sampling technique has extended its research into many ways, particularly in applications with Markov Chain Monte Carlo (MCMC). Given an ergodic Markov chain, the idea of perfect sampling is, in essence, to generate a sample from the exact stationary distribution in finites steps. Two methods for developing such a perfect samplingalgorithm are known, the one based on coupling from the past (CFTP) by Propp and Wilson (1996) and the other via rejection sampling by Fill (1998).
Fill's algorithm has been known for its advantage to be independent of the running time, but also for its intricacy in both theory and practice. The recent paper by Fill, Machida, Murdoch, and Rosenthal (1999) spelled out a general framework for his algorithm, and proposed various possible extension Hobert, Robert, and Tittering (1999) have studied CFTP algorithm for an MCMC evaluation as simple mixture problems. Their study was limited to two-and three-component mixtures, and suggested a difficulty in implementing CFTP algorithm when the mixture has more than three components. We demonstrate how a framework of Fill's algorithm can be applied for this particular mixture problem, and propose a perfect sampling algorithm using a cross-monotone chain. Our algorithm can be implemented with any number of components, but may not perform well as the number of components increases. The objective of this study is to further stimulate practical applications based on Fill's algorithm. Laird Breyer. A new coupling construction for Markov chains. July 10, 1999.
Abstract: Many branches of Probability Theory have benefited from coupling constructions, not least the theory of Markov chains. The coupling inequality for example allows us to bound the total variation distance of a Markov chain to its stationary distribution, in an entirely probabilistic way. This has led recently to computable bounds. As another example, Perfect Simulation exploits couplings to produce exact samples from distributions that are hard to simulate from. A central problem is the actual construction of the coupling, which tends to be problem specific and something of an art form. The "splitting technique" is one method which is very general and has been very successful for years. Among its drawbacks are the need to perform analytic estimates before the method can be applied. In this talk, we shall propose a "more automatic" way of coupling Markov chains, which does not require such estimates, and is applicable to a variety of chains. Kajsa Fröjd. Perfect simulation of some multitype spatial point processes. Jem Corcoran. Perfect Simulation with Applications to Statistical Mechanics. February 24, 2000.
Abstract: "Perfect simulation"-- Markov chain Monte Carlo with no statistical error. It sounds great, it even looks kind of easy. So why aren't you doing it? Probably because, as they say, "the devil is in the details." Perfect simulation methods are based on the idea of coupling sample paths of a Markov chain. In this talk we will explore the details of three specific examples, each having specific implementation difficulties. We will consider an unbounded storage model where paths must be coupled on a continuous state space, the standard Ising model of magnetism (both the ferromagnet and antiferomagnet) where paths are running around in a more complicated state space of "spin configurations", and the hard-core lattice gas model where sample paths will naturally repel one another despite the fact that our goal is to get them to meet. E. Olusegun George, Giri Narasimhan, and Anthony Quas. Fast and exact Gibbs sampler: an application of Propp and Wilson's CFTP algorithm.
Abstract: Gibbs Sampling is a popular technique for generating samples from posterior distributions with multidimensional parameters. Often the posterior distributions for such problems are analytically or numerically complex. As in the general MCMC procedure, the samples for posterior analysis are usually collected after a burn-out period, by which time, it is assumed that the sequence of generated values are close to those that would be generated by the true posterior distribution. The problem with this method is that, in order to be sure that sample for posterior analysis is a reasonable representation of the true posterior, a very large number of iterations is typically used for burnout. Furthermore, although the bias introduced by the choice of initial value may become vanishingly small with increasing number of iterations, it is never completely eliminated. In practice, the choice of number of iterations required for burnout is arbitrary and many researchers have proposed less than satisfactory techniques to approach the question. We introduce an alternative method to perform Gibbs Sampling that addresses the above-mentioned shortcomings. The practical advantages of all our methods are as follows: the samples generated are ‘perfect', i. e., the bias is completely eliminated; the method has a well-defined stopping rule and the expected number of iterations needed for burnout can be estimated in many cases. We illustrate our method by computing posterior distribution of in a teratology application. Our method is an extension of the techniques for perfect sampling suggested by Propp and Wilson (1996).
Surveys and Representative Articles from Related Areas.
Information on heuristic algorithms for assessing Markov chain convergence to the stationary distribution (among other things) can be found in these resources:
Alan D. Sokal. Monte Carlo methods in statistical mechanics: Foundations and new algorithms, 1989. Lecture notes from Cours de Troisième Cycle de la Physique en Suisse Romande. Updated in 1996 for the Cargèse Summer School on ``Functional Integration: Basics and Applications.'' (Click here to download.)
MCMC Preprint Service Surveys of rigorous bounds on the convergence rate of Markov chains:
Persi Diaconis and Laurent Saloff-Coste. What do we know about the Metropolis algorithm?, In Proceedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing , pages 112-129, 1995.
Mark Jerrum and Alistair Sinclair. The Markov chain Monte Carlo method: an approach to approximate counting and integration, in ``Approximation Algorithms for NP-hard Problems,'' D. S.Hochbaum ed., PWS Publishing, Boston, 1996. (Click here to download.) A good introduction to stopping times with many examples and references:
A more recent and more developed reference on stopping times:
Persi Diaconis and James A. Fill. Strong stationary times via a new form of duality. The Annals of Probability , 18:1483--1522, 1990. Representative articles on stochastically recursive sequences:
Gérard Letac. A contraction principle for certain Markov chains and its applications. In Random Matrices and Their Applications , volume 50 of Contemporary Mathematics , pages 263--273, 1986.
Reversible Markov chains:
David J. Aldous and James A. Fill. Reversible Markov Chains and Random Walks on Graphs. Book in preparation, stat. berkeley. edu/
Perguntas frequentes.
When explaining CFTP to an audience, there are invariably many questions. Included below are some of these questions together with their answers. [Adapted from section 1.4 of Wilson (2000), first published by the AMS.]
Q : If we always end up in state 3 no matter where we start, then how is that a random sample?
A : For a given particular sequence of coin flips, every possible starting state ends up in state 3. But for a different (random) sequence of coin flips, every possible starting state may end up in a different final state at time 0.
Q : What if we just run the Markov chain forward until coalescence?
A : Consider the toy example of the previous section. Coalescence only occurs at the states 0 and n , so the result would be very far from being distributed according to π.
Q : What happens if we use fresh coins rather than re-using the same coins?
A : Consider the toy example of the previous section, and set n =2 (so the states are 0,1,2). Then one can check that the probability of outputting state 1 has a binary expansion of 0.0010010101001010101010101001. which is neither rational nor very close to 1/3.
Q : Rather than doubling back in time, what if we double forward in time, and stop the Markov chain at the first power of 2 greater than or equal to the coalescence time?
A : As before, set n =2 in our toy example. One can check that the probability of outputting state 1 is 1/6 rather than 1/3.
Q : What if we instead .
A : Enough already! There do exist other ways to generate perfectly random samples, but the only obvious change that can be made to the CFTP algorithm without breaking it is changing the sequence of starting times in the past from powers of 2 to some other sequences integers.
Q : If coupling forwards in time doesn't work, then why does going backwards in time work?
A : If you did not like the first explanation, then another way to look at it is that a virtual Markov chain has been running for all time, and so today the state is random. If we can (with probability 1) figure out today's state by looking at some of the recent randomizing operations, then we have a random state.
Q : If going backwards in time works, then why doesn't going forwards in time work?
A1 : There is a way to obtain perfectly random samples by running forwards in time (Wilson 2000), but none of the obvious variations described above work.
A2 : CFTP determines the state at a deterministic time, any one of which is distributed according to π. The variations suggested above determine the state at a random time, where that time is correlated with the moves of the Markov chain in a complicated way, and these correlations mess things up.
Q : Why did you step back by powers of 2, when any other sequence would have worked?
A : For efficiency reasons. Let $T^*$ be the best time in the past at which to start, i. e. the smallest integer for which starting at time $-T^*$ leads to coalescence. By stepping back by powers of 2, the total number of Markov chain steps simulated is never larger than $4 T^*$ (and closer to $2.8 T^*$ ``on average''); see Propp and Wilson (1996), section 5.2. If we had stepped back by one each time, then the number of simulated steps would have been $ \approx (1/2) (T^*)^2$. If we had stepped back quadratically, then the number of simulated steps would have been about $(1/3) (T^*)^ $. For this reason some people prefer quadratic backoff when $T^*$ is fairly small.
Q : So you need the state space to be monotone?
A : That would certainly be very useful, but there are many counterexamples to the proposition that is necessary. See .
Q : Can you do CFTP on Banach spaces?
A : I don't know. Depends on whether or not you can figure out the state at time 0.
Q : Where's the proof of efficiency?
A : In the monotone setting, loosely speaking, CFTP is efficient whenever the Markov chain mixes rapidly; see . There also proofs of efficiency for certain non-monotone settings.
Q (?): But you still need to analyze the mixing time, since otherwise you won't know how long it will take.
A : Wrong. The principal advantage of using CFTP (or Fill's algorithm) is that you don't need these a priori mixing time bounds in order to run the algorithm, collect perfectly random samples, and carry on with the rest of the research project. The fact that these are perfectly random samples is icing on the cake. (But I still think that mixing time bounds are interesting.)
Q : But sampling spin glass configurations is NP-hard.
A : The spin-glass Markov chain that you're using is probably very slowly mixing in the worst case. CFTP will not be faster than the mixing time of the underlying Markov chain.
Q : Is CFTP like a ZPP or Las Vegas algorithm for random sampling?
A : ``Yes'' for Las Vegas, ``sometimes'' for ZPP. [A Las Vegas algorithm uses randomness to compute a deterministic function. The running time is a random variable, but with probability 1 it returns an answer, and when it does so, the answer is correct. A Monte Carlo algorithm in contrast does not guarantee a correct answer. A problem is in ZPP if it can be solved by polynomial expected time Las Vegas algorithm.]
Q : Could you make a hybrid algorithm, which starts out by doing CFTP, but then does something different if CFTP starts to take a long time?
A : Yes. This would be like experiment $C_T$.
Q : What's the probability that CFTP takes a long time?
A : The tail distribution of the running time decays geometrically. More can be said if the Markov chain is monotone, see .
Q : What if I don't want to wait for $10^ $ years. Does this make CFTP biased? Is it really better than forwards coupling?
A1 : The answer to the second question is yes. How long are you willing to wait? With forward coupling, that's how long you wait. With CFTP your average waiting time is probably much smaller.
A2 : The answer to the first question depends on what you mean by ``biased''. No-one disputes that the systematic bias is zero. The so-called ``user-impatience bias'' (the effect of an impatient user interrupting a simulation and progress) is a second order effect that pertains to most random sampling algorithms that most people would not call biased.
A3 : If you're genuinely concerned about the quality of your random samples, you should first spend time picking a good pseudo-random number generator.
A4 : Anyone concerned about ``user-impatience bias'' should be equally concerned about ``user-patience bias'': if an experimenter run simulations until some deadline, and then lets the last one finish before quitting, then the resulting collection of samples is biased to contain more samples that take a long time to generate. But if the experimenter instead aborts the last simulation (unless there are no samples so far, in which case (s)he lets it finish), then the resulting collection of samples is unbiased. See Glynn and Heidelberger.
Q : Can you quantify the user-impatience bias?
A : If you generate N samples before your deadline, and then average some function of these samples, the bias is at most Pr[N=0]. If you do something sensible in the event that N=0, the bias will be even less. See Glynn and Heidelberger.
Gallery of Pictures and Java Applets Relevant to the Articles.
Postscript illustration of the Green-Murdoch random walk Metropolis bisection coupling.
The 4-state Potts model is particularly interesting, click here to learn why and to see much larger perfectly random 4-state Potts configurations.
Perfect simulations of the area-interaction point process (from Kendall's web page of abstracts):
(a) attractive case and (b) repulsive case.
Definitions of Selected Terms.
Included here are the definitions of some terms used but not defined above.
area-interaction point process : Defined relative to the Poisson point process, the probability density of a configuration is given an extra weight that is exponentially small or large in the area of the union of circles (or other fixed shapes) centered about the points. When the extra weight is exponentially small (large) in the area, in a configuration with many points, the points will tend to be nearby (far away), giving the attractive (repulsive) area-interaction point process.
continuum random cluster model : Defined relative to a Poisson point process, clusters are formed by taking the union of circles (or other shapes) centered about the points. The probability density of a configuration gets an extra factor of q for each cluster that it contains. The discrete version of the continuum random cluster model is not the random cluster model, but rather is related to it in the same way that site percolation is related to bond percolation.
ice model : An ice configuration is an Eulerian orientation of a graph, i. e. an assignment of directions to the edges of a graph so that at each vertex the number of edges directed towards it equals the number of edges directed away from it. Some edges may be fixed at the boundaries of the graph, but given those constraints, all configurations are equally likely. (The term ``ice'' comes from an analogy with ice, where the vertices correspond to water molecules, the edges to hydrogen bonds, and the edge orientations determine which water molecule's hydrogren atom is involved in the hydrogen bond.)
Potts model : Each vertex on a graph (typically a grid) is assigned one of q different colors. The probability of a configu - ration gets an extra weight for each edge whose two endpoints are the same color. (The weight is thought of as being related to an interaction energy and the temperature.) When the weights are at least one, the model is ``ferromagnetic'', if they are less than one, then the model is ``antiferromagnetic''. The case q =2 is the Ising model. When the weight factor is zero, one obtains proper q - colorings.
random-cluster model : Also known as the Fortuin--Kasteleyn random cluster model, and bond-correlated percolation, it can be defined relative to ordinary bond percolation (each edge occuring independently with probability p ), where the probability of a configuration gets an extra weight of q for each connected component it contains. When q is a positive integer, this model is closely related to the q - state ferromagnetic Potts model. When q →0 and then p →0, the result is a random spanning tree. When q=1 one gets ordinary percolation.
spanning tree : A connected, acyclic subgraph of a given graph. If the graph is directed, then ``spanning tree'' refers to an ``arborescence'', in which a distinguished vertex is the ``root'' of the tree, and the edges of the tree are directed to - wards the root. A random spanning tree is such a tree (or arborescence) chosen uniformly at random, or if the graph is weighted, with probability proportional to the products of the weights of the edges contained in the tree.
Widom-Rowlinson model : A model in which there are typically two (but possibly more) types of particles, where particles of different types can't be nearby. In the discrete case, there is a graph, each vertex may contain zero or one particles, and no edge has two different types of particles on its endpoints. The probability of a legal configuration is proportional to some constant (the ``fugacity'' or ``activity'') raised to the power of the number of particles. In the continuous case, particles lie in the plane (or other manifold), and particles of different type must be some minimum distance apart. The probability density of a configuration is what one would get by randomly assigning particle types to the points of a Poisson process, and conditioning on the configuration being legal.
Perfect Sampling Code on the Web.
There seems to be enough perfect sampling code on the web to warrant making a section of links to it. Here is a partial listing, I will add more links as I track them down or they are called to my attention.