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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 ..."
Abstract

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...
Geometric Ergodicity of Gibbs and Block Gibbs Samplers for a Hierarchical Random Effects Model
, 1998
"... We consider fixed scan Gibbs and block Gibbs samplers for a Bayesian hierarchical random effects model with proper conjugate priors. A drift condition given in Meyn and Tweedie (1993, Chapter 15) is used to show that these Markov chains are geometrically ergodic. Showing that a Gibbs sampler is geom ..."
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Cited by 33 (8 self)
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We consider fixed scan Gibbs and block Gibbs samplers for a Bayesian hierarchical random effects model with proper conjugate priors. A drift condition given in Meyn and Tweedie (1993, Chapter 15) is used to show that these Markov chains are geometrically ergodic. Showing that a Gibbs sampler is geometrically ergodic is the first step towards establishing central limit theorems, which can be used to approximate the error associated with Monte Carlo estimates of posterior quantities of interest. Thus, our results will be of practical interest to researchers using these Gibbs samplers for Bayesian data analysis. Key words and phrases: Bayesian model, Central limit theorem, Drift condition, Markov chain, Monte Carlo, Rate of convergence, Variance Components AMS 1991 subject classifications: Primary 60J27, secondary 62F15 1 Introduction Gelfand and Smith (1990, Section 3.4) introduced the Gibbs sampler for the hierarchical oneway random effects model with proper conjugate priors. Rosen...
Hierarchical Models: A Current Computational Perspective
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2000
"... Hierarchical models (HMs) provide a flexible framework for modeling data. The ongoing development of techniques like the EM algorithm and Markov chain Monte Carlo has enabled statisticians to make use of increasingly more complicated HMs over the last few decades. In this article, we consider Bay ..."
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Cited by 9 (1 self)
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Hierarchical models (HMs) provide a flexible framework for modeling data. The ongoing development of techniques like the EM algorithm and Markov chain Monte Carlo has enabled statisticians to make use of increasingly more complicated HMs over the last few decades. In this article, we consider Bayesian and frequentist versions of a general, twostage HM, and describe several examples from the literature that illustrate its versatility. Some key aspects of the computational techniques that are currently used in conjunction with this HM are then examined in the context of McCullagh and Nelder's (1989) salamander data. Several areas that are ripe for new research are identified.
Discussion of "The Art of Data Augmentation" by van Dyk and Meng
, 2000
"... that Y 1 ; : : : ; Y n are iid from a Student's t distribution with degrees of freedom, location parameter and scale parameter . Denote this distribution by t(; ; ). Suppose that is known and let (; ) be a prior that yields a proper posterior given by (; jy) = f(yj; )(; ) m(y) (1) where y = (y ..."
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that Y 1 ; : : : ; Y n are iid from a Student's t distribution with degrees of freedom, location parameter and scale parameter . Denote this distribution by t(; ; ). Suppose that is known and let (; ) be a prior that yields a proper posterior given by (; jy) = f(yj; )(; ) m(y) (1) where y = (y 1 ; : : : ; y n ) T and m(y) is the marginal density of the data. The goal is to sample from this posterior. vD&M note that if it is assumed that Y 1 jq 1 <F13.1