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67
Markov chains for exploring posterior distributions
 Annals of Statistics
, 1994
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Cited by 751 (6 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review
 Journal of the American Statistical Association
, 1996
"... A critical issue for users of Markov Chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest. Research into methods of computing theoretical convergence bounds holds promise ..."
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Cited by 223 (6 self)
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A critical issue for users of Markov Chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest. Research into methods of computing theoretical convergence bounds holds promise for the future but currently has yielded relatively little that is of practical use in applied work. Consequently, most MCMC users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. After giving a brief overview of the area, we provide an expository review of thirteen convergence diagnostics, describing the theoretical basis and practical implementation of each. We then compare their performance in two simple models and conclude that all the methods can fail to detect the sorts of convergence failure they were designed to identify. We thus recommend a combination of strategies aimed at evaluating and accelerating MCMC sampler conver...
Regeneration in Markov Chain Samplers
, 1994
"... Markov chain sampling has received considerable attention in the recent literature, in particular in the context of Bayesian computation and maximum likelihood estimation. This paper discusses the use of Markov chain splitting, originally developed as a tool for the theoretical analysis of general s ..."
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Cited by 87 (5 self)
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Markov chain sampling has received considerable attention in the recent literature, in particular in the context of Bayesian computation and maximum likelihood estimation. This paper discusses the use of Markov chain splitting, originally developed as a tool for the theoretical analysis of general state space Markov chains, to introduce regeneration times into Markov chain samplers. This allows the use of regenerative methods for analyzing the output of these samplers, and can also provide a useful diagnostic of the performance of the samplers. The general approach is applied to several different samplers and is illustrated in a number of examples. 1 Introduction In Markov chain Monte Carlo, a distribution ß is examined by obtaining sample paths from a Markov chain constructed to have equilibrium distribution ß. This approach was introduced by Metropolis et al. (1953) and has recently received considerable attention as a method for examining posterior distributions in Bayesian infer...
Bayesian Methods for Hidden Markov Models  Recursive Computing in the 21st Century
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2002
"... Markov chain Monte Carlo (MCMC) sampling strategies can be used to simulate hidden Markov model (HMM) parameters from their posterior distribution given observed data. Some MCMC methods (for computing likelihood, conditional probabilities of hidden states, and the most likely sequence of states) use ..."
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Cited by 86 (8 self)
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Markov chain Monte Carlo (MCMC) sampling strategies can be used to simulate hidden Markov model (HMM) parameters from their posterior distribution given observed data. Some MCMC methods (for computing likelihood, conditional probabilities of hidden states, and the most likely sequence of states) used in practice can be improved by incorporating established recursive algorithms. The most important is a set of forwardbackward recursions calculating conditional distributions of the hidden states given observed data and model parameters. We show how to use the recursive algorithms in an MCMC context and demonstrate mathematical and empirical results showing a Gibbs sampler using the forwardbackward recursions mixes more rapidly than another sampler often used for HMM's. We introduce an augmented variables technique for obtaining unique state labels in HMM's and finite mixture models. We show how recursive computing allows statistically efficient use of MCMC output when estimating the hidden states. We directly calculate the posterior distribution of the hidden chain's state space size by MCMC, circumventing asymptotic arguments underlying the Bayesian information criterion, which is shown to be inappropriate for a frequently analyzed data set in the HMM literature. The use of loglikelihood for assessing MCMC convergence is illustrated, and posterior predictive checks are used to investigate application specific questions of model adequacy.
The practical implementation of Bayesian model selection
 Institute of Mathematical Statistics
, 2001
"... In principle, the Bayesian approach to model selection is straightforward. Prior probability distributions are used to describe the uncertainty surrounding all unknowns. After observing the data, the posterior distribution provides a coherent post data summary of the remaining uncertainty which is r ..."
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Cited by 85 (3 self)
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In principle, the Bayesian approach to model selection is straightforward. Prior probability distributions are used to describe the uncertainty surrounding all unknowns. After observing the data, the posterior distribution provides a coherent post data summary of the remaining uncertainty which is relevant for model selection. However, the practical implementation of this approach often requires carefully tailored priors and novel posterior calculation methods. In this article, we illustrate some of the fundamental practical issues that arise for two different model selection problems: the variable selection problem for the linear model and the CART model selection problem.
On the Convergence of Monte Carlo Maximum Likelihood Calculations
 Journal of the Royal Statistical Society B
, 1992
"... Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family f h ` : ` 2 \Theta g of nonnegative integrable functions, maximum likelihood estimates in the family obtained by normalizing the the ..."
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Cited by 59 (3 self)
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Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family f h ` : ` 2 \Theta g of nonnegative integrable functions, maximum likelihood estimates in the family obtained by normalizing the the functions to integrate to one can be approximated by Monte Carlo, the only regularity conditions being a compactification of the parameter space such that the the evaluation maps ` 7! h ` (x) remain continuous. Then with probability one the Monte Carlo approximant to the log likelihood hypoconverges to the exact log likelihood, its maximizer converges to the exact maximum likelihood estimate, approximations to profile likelihoods hypoconverge to the exact profile, and level sets of the approximate likelihood (support regions) converge to the exact sets (in Painlev'eKuratowski set convergence). The same results hold when there are missing data (Thompson and Guo, 1991, Gelfand and Carlin, 19...
Markov Chain Monte Carlo in Conditionally Gaussian State Space Models
 Biometrika
, 1996
"... Introduction Linear Gaussian state space models are used extensively, with unknown parameters usually estimated by maximum likelihood: Wecker & Ansley (1983), Harvey (1989). However, many time series and nonparametric regression applications, such as change point problems, outlier detection and swit ..."
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Cited by 54 (3 self)
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Introduction Linear Gaussian state space models are used extensively, with unknown parameters usually estimated by maximum likelihood: Wecker & Ansley (1983), Harvey (1989). However, many time series and nonparametric regression applications, such as change point problems, outlier detection and switching regression, require the full generality of the conditionally Gaussian model: Harrison & Stevens (1976), Shumway & Stoffer (1991), West & Harrison (1989), Gordon & Smith (1990). The presence of a large number of indicator variables makes it difficult to estimate conditionally Gaussian models using maximum likelihood, and a Bayesian approach using Markov chain Monte Carlo appears more tractable. We propose a new sampler, which is used to estimate an unknown function nonparametrically when there are jumps in the function and outliers in the observations; it is also applied to a time series change point problem previously discussed by Gordon & Smith (1990). For the first example th
Blocking Gibbs Sampling in Very Large Probabilistic Expert Systems
 Internat. J. Human–Computer Studies
, 1995
"... We introduce a methodology for performing approximate computations in very complex probabilistic systems (e.g. huge pedigrees). Our approach, called blocking Gibbs, combines exact local computations with Gibbs sampling in a way that complements the strengths of both. The methodology is illustrate ..."
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Cited by 46 (0 self)
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We introduce a methodology for performing approximate computations in very complex probabilistic systems (e.g. huge pedigrees). Our approach, called blocking Gibbs, combines exact local computations with Gibbs sampling in a way that complements the strengths of both. The methodology is illustrated on a realworld problem involving a heavily inbred pedigree containing 20;000 individuals. We present results showing that blockingGibbs sampling converges much faster than plain Gibbs sampling for very complex problems.
On the Markov chain central limit theorem. Probability Surveys
, 2004
"... The goal of this mainly expository paper is to describe conditions which guarantee a central limit theorem for functionals of general state space Markov chains with a view towards Markov chain Monte Carlo settings. Thus the focus is on the connections between drift and mixing conditions and their im ..."
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Cited by 46 (11 self)
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The goal of this mainly expository paper is to describe conditions which guarantee a central limit theorem for functionals of general state space Markov chains with a view towards Markov chain Monte Carlo settings. Thus the focus is on the connections between drift and mixing conditions and their implications. In particular, we consider three commonly cited central limit theorems and discuss their relationship to classical results for mixing processes. Several motivating examples are given which range from toy onedimensional settings to complicated settings encountered in Markov chain Monte Carlo. 1
Renewal theory and computable convergence rates for geometrically ergodic Markov chains
, 2003
"... We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are different from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start f ..."
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Cited by 41 (0 self)
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We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are different from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start from essentially the same assumptions of a drift condition toward a “small set. ” The estimates show a noticeable improvement on existing results if the Markov chain is reversible with respect to its stationary distribution, and especially so if the chain is also positive. The method of proof uses the firstentrance– lastexit decomposition, together with new quantitative versions of a result of Kendall from discrete renewal theory. 1. Introduction. Let {Xn:n ≥ 0