Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review
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Cites background from "Markov Chain Monte Carlo Convergenc..."
...Cowles and Carlin (1996) provide a description of the various tests and diagnostics that have been proposed....
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Cites background or methods from "Markov Chain Monte Carlo Convergenc..."
...See Brooks and Roberts (in press) and Cowles and Carlin (1996) for reviews of commonly used techniques of convergence assessment....
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...See Brooks and Roberts (in press) and Cowles and Carlin (1996) for reviews of commonly used techniques of convergence assessment. Gelman and Rubin (1992a,b) pointed out that, in many problems, lack of convergence can be easily determined from multiple independent sequences but cannot be diagnosed using simulation output from any single sequence. They proposed a method using multiple replications of the chain to decide whether or not stationarity has been achieved within the second half of each of the sample paths. The idea behind this is an implicit assumption that convergence will have been achieved within the first half of the sample paths, and the validity of this assumption is essentially being tested by the diagnostic. This method's popularity may be largely due to its implementational simplicity and the fact that generic code is widely available to implement the method. In addition, even when this method is not formally used, something like it is often done implicitly or informally. For example, Green, Roesch, Smith, and Strawderman (1994) wrote, in implementing a Gibbs sampler, "In the present study we used . . . one long chain. Experimentation with different starting values convinced us that the chain was converging and covering the entire posterior distribution." As is often the case in statistics, it generally useful to study the formal procedures that lie behind an informal method, in this case for monitoring the mixing of multiple sequences. In this article, we generalize the method of Gelman and Rubin (1992a) by (1) adding graphical methods for tracking the approach to convergence; (2) generalizing the scale reduction factor to track measures of scale other than the variance; and (3) extending to multivariate summaries....
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...See Brooks and Roberts (in press) and Cowles and Carlin (1996) for reviews of commonly used techniques of convergence assessment. Gelman and Rubin (1992a,b) pointed out that, in many problems, lack of convergence can be easily determined from multiple independent sequences but cannot be diagnosed using simulation output from any single sequence. They proposed a method using multiple replications of the chain to decide whether or not stationarity has been achieved within the second half of each of the sample paths. The idea behind this is an implicit assumption that convergence will have been achieved within the first half of the sample paths, and the validity of this assumption is essentially being tested by the diagnostic. This method's popularity may be largely due to its implementational simplicity and the fact that generic code is widely available to implement the method. In addition, even when this method is not formally used, something like it is often done implicitly or informally. For example, Green, Roesch, Smith, and Strawderman (1994) wrote, in implementing a Gibbs sampler, "In the present study we used ....
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2,938 citations
References
35,161 citations
"Markov Chain Monte Carlo Convergenc..." refers methods in this paper
...In a surprisingly short period of time, Markov chain Monte Carlo (MCMC) integration methods, especially the Metropolis-Hastings algorithm (Metropolis et al., 1953; Hastings, 1970) and the Gibbs sampler (Geman and Geman, 1984; Gelfand and Smith, 1990) have emerged as extremely popular tools for the analysis of complex statistical models....
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18,761 citations
"Markov Chain Monte Carlo Convergenc..." refers methods in this paper
...pler ( Geman and Geman, 1984; Gelfand and Smith, 1990) have emerged as extremely popular tools for...
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...…integration methods, especially the MetropolisHastings algorithm (Hastings 1970; Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller 1953) and the Gibbs sampler (Geman and Geman 1984; Gelfand and Smith 1990) have emerged as extremely popular tools for the analysis of complex statistical models....
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14,965 citations
"Markov Chain Monte Carlo Convergenc..." refers methods in this paper
...In a surprisingly short period, Markov chain Monte Carlo (MCMC) integration methods, especially the MetropolisHastings algorithm (Hastings 1970; Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller 1953) and the Gibbs sampler (Geman and Geman 1984; Gelfand and Smith 1990) have emerged as…...
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...pecially the Metropolis-Hastings algorithm (Metropolis et al., 1953; Hastings, 1970 ) and the Gibbs sam-...
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13,884 citations
"Markov Chain Monte Carlo Convergenc..." refers background or methods in this paper
...The convergence diagnostics of Gelman and Rubin (1992) and of Raftery and Lewis (1992) currently are the most popular amongst the statistical community, at least in part because computer programs for their implementation are available from their creators. In addition to these two, we discuss the methods of Geweke (1992), Roberts (1992, 1994), Ritter and Tanner (1992), Zellner and Min (1995), Liu, Liu, and Rubin (1992), Garren and Smith (1993), Johnson (1994), Mykland, Tiermey, and Yu (1995), Yu (1994), and Yu and Mykland (1994)....
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...…or Univariate/ Ease Quantitative multiple full joint Bias! of Method graphical chains Theoretical basis distribution variance Applicability use Gelman and Rubin (1992) Quantitative Multiple Large-sample normal theory Univariate Bias Any MCMC a Raftery and Lewis (1992) Quantitative Single…...
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...The convergence diagnostics of Gelman and Rubin (1992) and of Raftery and Lewis (1992) currently are the most popular in the statistical community, at least in part because computer programs for their implementation are available from their creators....
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...The convergence diagnostics of Gelman and Rubin (1992) and of Raftery and Lewis (1992) currently are the most popular amongst the statistical community, at least in part because computer programs for their implementation are available from their creators....
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10,225 citations
"Markov Chain Monte Carlo Convergenc..." refers background in this paper
...The plotted lines are "lowess" smooths (Cleveland 1979) of log-transformed In and Dn values....
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