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Perfect Gibbs sampling is a method to turn Markov Chain Monte Carlo (MCMC) samplers into exact generators for independent random vectors. We show that a perfect Gibbs sampling algorithm suggested in the literature is not always generating from the correct distribution. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Perfect slice sampling is a method to turn Markov Chain Monte Carlo (MCMC) samplers into exact generators for independent random variates. We show that the simplest version of the perfect slice sampler suggested in the literature does not always sample from the target distribution. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Perfect simulation  and the EM algorithm are the main topics in this thesis. In paper I, we present coupling from the past (CFTP) algorithms that generate perfectly distributed samples from the multi-type Widom--Rowlin-son (W--R) model and some generalizations of it. The classical W--R model is a point process in the plane or the  space consisting of points of several different types. Points of different types are not allowed to be closer than some specified distance, whereas points of the same type can be arbitrary close. A stick-model and soft-core generalizations are also considered. Further, we  generate samples without edge effects, and give a bound on sufficiently small intensities (of the points) for the algorithm to terminate. In paper II, we consider the  forestry problem on how to estimate  seedling dispersal distributions and effective plant fecundities from spatially data of adult trees  and seedlings, when the origin of the seedlings are unknown.   Traditional models for fecundities build on allometric assumptions, where the fecundity is related to some  characteristic of the adult tree (e.g.\ diameter). However, the allometric assumptions are generally too restrictive and lead to nonrealistic estimates. Therefore we present a new model, the unrestricted fecundity (UF) model, which uses no allometric assumptions. We propose an EM algorithm to estimate the unknown parameters.   Evaluations on real and simulated data indicates better performance for the UF model. In paper III, we propose  EM algorithms to  estimate the passage time distribution on a graph.Data is obtained by observing a flow only at the nodes -- what happens on the edges is unknown. Therefore the sample of passage times, i.e. the times it takes for the flow to stream between two neighbors, consists of right censored and uncensored observations where it sometimes is unknown which is which.       For discrete passage time distributions, we show that the maximum likelihood (ML) estimate is strongly consistent under certain  weak conditions. We also show that our propsed EM algorithm  converges to the ML estimate if the sample size is sufficiently large and the starting value is sufficiently close to the true parameter. In a special case we show that it always converges.  In the continuous case, we propose an EM algorithm for fitting  phase-type distributions to data.

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. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

In this thesis we study two probabilistic models defined on graphs: the Stochastic Block model and the Exponential Random Graph. Therefore, this thesis is divided in two parts. In the first part, we introduce the Krichevsky-Trofimov estimator for the number of communities in the Stochastic Block Model and prove its eventual almost sure convergence to the underlying number of communities, without assuming a known upper bound on that quantity. In the second part of this thesis we address the perfect simulation problem for the Exponential random graph model. We propose an algorithm based on the Coupling From The Past algorithm using a Glauber dynamics. This algorithm is efficient in the case of monotone models. We prove that this is the case for a subset of the parametric space. We also propose an algorithm based on the Backward and Forward algorithm that can be applied for monotone and non monotone models. We prove the existence of an upper bound for the expected running time of both algorithms. / Nessa tese estudamos dois modelos probabilísticos definidos em grafos: o modelo estocástico por blocos e o modelo de grafos exponenciais. Dessa forma, essa tese está dividida em duas partes. Na primeira parte nós propomos um estimador penalizado baseado na mistura de Krichevsky-Trofimov para o número de comunidades do modelo estocástico por blocos e provamos sua convergência quase certa sem considerar um limitante conhecido para o número de comunidades. Na segunda parte dessa tese nós abordamos o problema de simulação perfeita para o modelo de grafos aleatórios Exponenciais. Nós propomos um algoritmo de simulação perfeita baseado no algoritmo Coupling From the Past usando a dinâmica de Glauber. Esse algoritmo é eficiente apenas no caso em que o modelo é monotóno e nós provamos que esse é o caso para um subconjunto do espaço paramétrico. Nós também propomos um algoritmo de simulação perfeita baseado no algoritmo Backward and Forward que pode ser aplicado à modelos monótonos e não monótonos. Nós provamos a existência de um limitante superior para o número esperado de passos de ambos os algoritmos.

In this thesis we study two probabilistic models defined on graphs: the Stochastic Block model and the Exponential Random Graph. Therefore, this thesis is divided in two parts. In the first part, we introduce the Krichevsky-Trofimov estimator for the number of communities in the Stochastic Block Model and prove its eventual almost sure convergence to the underlying number of communities, without assuming a known upper bound on that quantity. In the second part of this thesis we address the perfect simulation problem for the Exponential random graph model. We propose an algorithm based on the Coupling From The Past algorithm using a Glauber dynamics. This algorithm is efficient in the case of monotone models. We prove that this is the case for a subset of the parametric space. We also propose an algorithm based on the Backward and Forward algorithm that can be applied for monotone and non monotone models. We prove the existence of an upper bound for the expected running time of both algorithms. / Nessa tese estudamos dois modelos probabilísticos definidos em grafos: o modelo estocástico por blocos e o modelo de grafos exponenciais. Dessa forma, essa tese está dividida em duas partes. Na primeira parte nós propomos um estimador penalizado baseado na mistura de Krichevsky-Trofimov para o número de comunidades do modelo estocástico por blocos e provamos sua convergência quase certa sem considerar um limitante conhecido para o número de comunidades. Na segunda parte dessa tese nós abordamos o problema de simulação perfeita para o modelo de grafos aleatórios Exponenciais. Nós propomos um algoritmo de simulação perfeita baseado no algoritmo Coupling From the Past usando a dinâmica de Glauber. Esse algoritmo é eficiente apenas no caso em que o modelo é monotóno e nós provamos que esse é o caso para um subconjunto do espaço paramétrico. Nós também propomos um algoritmo de simulação perfeita baseado no algoritmo Backward and Forward que pode ser aplicado à modelos monótonos e não monótonos. Nós provamos a existência de um limitante superior para o número esperado de passos de ambos os algoritmos.

Perfect simulations are methods, which convert suitable Markov chain Monte Carlo (MCMC) algorithms into algorithms which return exact draws from the target distribution, instead of approximations based on long-time convergence to equilibrium. In recent years a lot of various perfect simulation algorithms were developed. This work provides a unified exposition of some perfect simulation algorithms with applications to spatial point processes, especially to the Strauss process and area-interaction process. Described algorithms and their properties are compared theoretically and also by a simulation study.