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Markov chain monte carlo convergence diagnostics
 JASA
, 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 367 (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 convergence, including applying diagnostic procedures to a small number of parallel chains, monitoring autocorrelations and crosscorrelations, and modifying parameterizations or sampling algorithms appropriately. We emphasize, however, that it is not possible to say with certainty that a finite sample from an MCMC algorithm is representative of an underlying stationary distribution. 1
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 ..."
Abstract

Cited by 22 (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
, 1996
"... ..."
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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Cited by 1 (0 self)
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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.
Computer Based Statistical Treatment in Models with Incidental Parameters Inspired by Car Crash Data
"... in recent years. We study computer intensive methods that can be used in complex situations where it is not possible to express the likelihood estimates or the posterior analytically. The work is inspired by a set of car crash data from real traffic. We formulate and develop a model for car crash da ..."
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in recent years. We study computer intensive methods that can be used in complex situations where it is not possible to express the likelihood estimates or the posterior analytically. The work is inspired by a set of car crash data from real traffic. We formulate and develop a model for car crash data that aims to estimate and compare the relative collision safety among different car models. This model works sufficiently well, although complications arise due to a growing vector of incidental parameters. The bootstrap is shown to be a useful tool for studying uncertainties of the estimates of the structural parameters. This model is further extended to include driver characteristics. In a Poisson model with similar, but simpler structure, estimates of the structural parameter in the presence of incidental parameters are studied. The profile likelihood, bootstrap and the delta method are compared for deterministic and random incidental parameters. The same asymptotic properties, up to first order, are seen for deterministic as well as random
Abstract Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review
"... 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
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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 convergence, including applying diagnostic procedures to a small number of parallel chains, monitoring autocorrelations and crosscorrelations, and modifying parameterizations or sampling algorithms appropriately. We emphasize, however, that it is not possible to say with certainty that a finite sample from an MCMC algorithm is representative of an underlying stationary distribution.
Diagnostics: A Comparative Review
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
Abstract
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
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...
AN INVESTIGATION OF A BAYESIAN DECISIONTHEORETIC PROCEDURE IN THE CONTEXT OF MASTERY TESTS
, 2007
"... procedure in the context of mastery tests ..."
Is PartialDimension Convergence a Problem for MCMC Algorithms?
"... Bayesian applications appear increasingly often in political science research along with the concomitant Markov chain Monte Carlo estimation/marginalization process often required to get working inferences from complicated joint posterior distributions. Recent examples include: Gill and Walker (2005 ..."
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Bayesian applications appear increasingly often in political science research along with the concomitant Markov chain Monte Carlo estimation/marginalization process often required to get working inferences from complicated joint posterior distributions. Recent examples include: Gill and Walker (2005), Hill and