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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 220 (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...
Image Sequence Restoration Using Gibbs Distributions
, 1995
"... This thesis addresses a number of issues concerned with the restoration of one type of image sequence, namely archived black and white motion pictures. These are often a valuable historical record, but because of the physical nature of the film they can suffer from a variety of degradations which re ..."
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Cited by 22 (0 self)
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This thesis addresses a number of issues concerned with the restoration of one type of image sequence, namely archived black and white motion pictures. These are often a valuable historical record, but because of the physical nature of the film they can suffer from a variety of degradations which reduce their usefulness. The main visual defects are `dirt and sparkle' due to dust and dirt becoming attached to the film, or abrasion removing the emulsion, and `line scratches' due to the film running against foreign bodies in the camera or projector. For an image
Looking at Markov Samplers through Cusum Path Plots: a simple diagnostic idea
, 1994
"... In this paper, we propose to monitor a Markov chain sampler using the cusum path plot of a chosen 1dimensional summary statistic. We argue that the cusum path plot can bring out, more effectively than the sequential plot, those aspects of a Markov sampler which tell the user how quickly or slowly t ..."
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Cited by 12 (3 self)
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In this paper, we propose to monitor a Markov chain sampler using the cusum path plot of a chosen 1dimensional summary statistic. We argue that the cusum path plot can bring out, more effectively than the sequential plot, those aspects of a Markov sampler which tell the user how quickly or slowly the sampler is moving around in its sample space, in the direction of the summary statistic. The proposal is then illustrated in four examples which represent situations where the cusum path plot works well and not well. Moreover, a rigorous analysis is given for one of the examples. We conclude that the cusum path plot is an effective tool for convergence diagnostics of a Markov sampler and for comparing different Markov samplers. KEY WORDS: Convergence diagnostic; Cusum path plot, Markov sampler; Mixing; Sequential plot; Summary statistic. Research supported in part by ARO Grant DAAL0391G007. y Research supported in part by NSF Grant DMS9305601. 1 Introduction As Markov chain Mon...
Comment: Extracting more diagnostic information from a single run using Cusum Path Plot
 Statistical Science
"... Bin Yu is Assistant Professor, Department of Statistics, University of California, Berkeley, CA 947203860. Research supported in part by Grant DMS9322817 from the National Science Foundation and Grant DAAH0494G 0232 from the Army Research Office. Tanner, Sinhua, and Hall (1992) suggested dia ..."
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Cited by 1 (0 self)
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Bin Yu is Assistant Professor, Department of Statistics, University of California, Berkeley, CA 947203860. Research supported in part by Grant DMS9322817 from the National Science Foundation and Grant DAAH0494G 0232 from the Army Research Office. Tanner, Sinhua, and Hall (1992) suggested diagnostic statistics based on importance weights, using either multiple chains or a single chain. A priori bounds on the convergence rate can be found in Rosenthal (1993), and Mengersen and Tweedie (1993), but unfortunately these theoretical bounds are currently known only in some very special cases. For other references on existing diagnostic tools, see the recent and thorough review by Cowles (1994). On the other hand, Yu and Mykland (1994) suggest that more information can be extracted from a single run than previously believed. The device is the cusum path plot, which brings out the local mixing behavior of the Markov chain in the direction of a chosen 1dim summar
Estimating L¹ Error of Kernel Estimator: Monitoring Convergence of Markov Samplers
"... In many Markov chain Monte Carlo problems, the target density function is known up to a normalization constant. In this paper, we take advantage of this knowledge to facilitate the convergence diagnostic of a Markov sampler by estimating the L 1 error of a kernel estimator. Firstly, we propose an ..."
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In many Markov chain Monte Carlo problems, the target density function is known up to a normalization constant. In this paper, we take advantage of this knowledge to facilitate the convergence diagnostic of a Markov sampler by estimating the L 1 error of a kernel estimator. Firstly, we propose an estimator of the normalization constant which is shown to be asymptotically normal under mixing and moment conditions. Secondly, the L 1 error of the kernel estimator is estimated using the normalization constant estimator, and the ratio of the estimated L 1 error to the true L 1 error is shown to converge to 1 in probability under similar conditions. Thirdly, we propose a sequential plot of the estimated L 1 error as a tool to monitor the convergence of the Markov sampler. Finally, a 2dimensional bimodal example is given to illustrate the proposal, and two Markov samplers are compared in the example using the proposed diagnostic plot. KEY WORDS: fimixing; Diagnostic; Normalization...
Inference and Monitoring Convergence (chapter for Gilks, Richardson, and Spiegelhalter book)
"... this article we present yet another example, from our current applied research. Figure 0.1 displays an example of slow convergence from a Markov chain simulation for a hierarchical Bayesian model for a pharmacokinetics problem (see Bois et al., 1994, for details). The simulations were done using a M ..."
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this article we present yet another example, from our current applied research. Figure 0.1 displays an example of slow convergence from a Markov chain simulation for a hierarchical Bayesian model for a pharmacokinetics problem (see Bois et al., 1994, for details). The simulations were done using a Metropolisapproximate Gibbs sampler (as in Section 4.4 of Gelman, 1992); due to the complexity of the model, each iteration was expensive in computer time, and it was desirable to keep the simulation runs as short as possible. Figures 1a and 1b display time series plots for a single parameter in the posterior distribution in two independent simulations, each of length 1000. The simulations were run in parallel simultaneously on two workstations in a network. It is clear from the separation of the two sequences that, after 1000 iterations, the simulations are still far from convergence. However, either sequence alone looks perfectly well behaved.
Estimating L 1 Error of Kernel Estimator: Monitoring Convergence of Markov Samplers
"... In many Markov chain Monte Carlo problems, the target density function is known up to a normalization constant. In this paper, we takeadvantage of this knowledge to facilitate the convergence diagnostic of a Markov sampler by estimating the L1 error of a kernel estimator. Firstly, we propose an esti ..."
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In many Markov chain Monte Carlo problems, the target density function is known up to a normalization constant. In this paper, we takeadvantage of this knowledge to facilitate the convergence diagnostic of a Markov sampler by estimating the L1 error of a kernel estimator. Firstly, we propose an estimator of the normalization constant whichisshown to be asymptotically normal under mixing and moment conditions. Secondly,theL1 error of the kernel estimator is estimated using the normalization constant estimator, and the ratio of the estimated L1 error to the true L1 error is shown to converge to 1 in probability under similar conditions. Thirdly, we propose a sequential plot of the estimated L1 error as a tool to monitor the convergence of the Markov sampler. Finally, a 2dimensional bimodal example is given to illustrate the proposal, and two Markov samplers are compared in the example using the proposed diagnostic plot.
Diagnostics: A Comparative Review
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