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89
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...
CODA: Convergence Diagnosis and Output Analysis Software for Gibbs sampling output Version 0.30
, 1995
"... ing beta ... 200 valid values Abstracting alpha ... 200 valid values Abstracting sigma ... 200 valid values Reading Data file... Abstracting beta ... 200 valid values Abstracting alpha ... 200 valid values Abstracting sigma ... 200 valid values 10 Next, you will be prompted to specify which (if any ..."
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Cited by 85 (5 self)
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ing beta ... 200 valid values Abstracting alpha ... 200 valid values Abstracting sigma ... 200 valid values Reading Data file... Abstracting beta ... 200 valid values Abstracting alpha ... 200 valid values Abstracting sigma ... 200 valid values 10 Next, you will be prompted to specify which (if any) variables take values restricted to either the range (0, 1) or to the positive real line. CODA requires this information in order to correctly compute Gelman and Rubin (1992)'s convergence diagnostic for nonnormal variables (see x4.2), and to produce kernel density estimates within the appropriate range (see x3.1). Are any variables restricted to values between 0 and 1 (y/n) ? 1: For the line example, you should respond n to this question. The next prompt to appear is as follows: Are any variables restricted to all positive values (y/n) ? 1: For the line example, you should respond y to this question, which causes the following display to appear: Available variables: ++...
The Number of Iterations, Convergence Diagnostics and Generic Metropolis Algorithms
 In Practical Markov Chain Monte Carlo (W.R. Gilks, D.J. Spiegelhalter and
, 1995
"... Introduction In order to use Markov chain Monte Carlo, MCMC, it is necessary to determine how long the simulation needs to be run. It is also a good idea to discard a number of initial "burnin " simulations, since from an arbitrary starting point it would be unlikely that the initial simulations ca ..."
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Cited by 29 (3 self)
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Introduction In order to use Markov chain Monte Carlo, MCMC, it is necessary to determine how long the simulation needs to be run. It is also a good idea to discard a number of initial "burnin " simulations, since from an arbitrary starting point it would be unlikely that the initial simulations came from the stationary distribution intended for the Markov chain. Also, consecutive simulations from Markov chains are dependent, sometimes highly so. Since saving all simulations can require a large amount of storage, researchers using MCMC sometimes prefer saving only every third, fifth, tenth, etc. simulation, especially if the chain is highly dependent. This is sometimes referred to as thinning the chain. While neither burnin nor thinning are mandatory practices, they both reduce the amount of data saved from a MCMC run. In this chapter, we outline a way of determining in advance the number of iterations needed for a given level of precision in a MCMC algorithm.
A simulation approach to convergence rates for Markov chain Monte Carlo algorithms
 Stat. and Comput
, 1996
"... Markov chain Monte Carlo (MCMC) methods, including the Gibbs sampler and the MetropolisHastings algorithm, are very commonly used in Bayesian statistics for sampling from complicated, highdimensional posterior distributions. A continuing source of uncertainty is how long such a sampler must be run ..."
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Cited by 21 (10 self)
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Markov chain Monte Carlo (MCMC) methods, including the Gibbs sampler and the MetropolisHastings algorithm, are very commonly used in Bayesian statistics for sampling from complicated, highdimensional posterior distributions. A continuing source of uncertainty is how long such a sampler must be run in order to converge approximately to its target stationary distribution. Rosenthal (1995b) presents a method to compute rigorous theoretical upper bounds on the number of iterations required to achieve a specified degree of convergence in total variation distance by verifying drift and minorization conditions. We propose the use of auxiliary simulations to estimate the numerical values needed in Rosenthal's theorem. Our simulation method makes it possible to compute quantitative convergence bounds for models for which the requisite analytical computations would be prohibitively difficult or impossible. On the other hand, although our method appears to perform well in our example problems...
The Matrix StickBreaking Process for Flexible MultiTask Learning
"... In multitask learning our goal is to design regression or classification models for each of the tasks and appropriately share information between tasks. A Dirichlet process (DP) prior can be used to encourage task clustering. However, the DP prior does not allow local clustering of tasks with respe ..."
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Cited by 19 (3 self)
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In multitask learning our goal is to design regression or classification models for each of the tasks and appropriately share information between tasks. A Dirichlet process (DP) prior can be used to encourage task clustering. However, the DP prior does not allow local clustering of tasks with respect to a subset of the feature vector without making independence assumptions. Motivated by this problem, we develop a new multitasklearning prior, termed the matrix stickbreaking process (MSBP), which encourages crosstask sharing of data. However, the MSBP allows separate clustering and borrowing of information for the different feature components. This is important when tasks are more closely related for certain features than for others. Bayesian inference proceeds by a Gibbs sampling algorithm and the approach is illustrated using a simulated example and a multinational application. 1.
Implementation and Performance Issues in the Bayesian And Likelihood . . .
 COMPUTATIONAL STATISTICS
, 2000
"... ..."
A Bayesian Analysis of Simulation Algorithms for Inference in Belief Networks,
 Networks
, 1993
"... A belief network is a graphical representation of the underlying probabilistic relationships in a complex system. Belief networks have been employed as a representation of uncertain relationships in computerbased diagnostic systems. These diagnostic systems provide assistance by assigning likeli ..."
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Cited by 17 (3 self)
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A belief network is a graphical representation of the underlying probabilistic relationships in a complex system. Belief networks have been employed as a representation of uncertain relationships in computerbased diagnostic systems. These diagnostic systems provide assistance by assigning likelihoods to alternative explanatory hypotheses in response to a set of findings or observations. Approximation algorithms have been used to compute likelihoods of hypotheses in large networks. We analyze the performance of leading Monte Carlo approximation algorithms for computing posterior probabilities in belief networks. The analysis differs from earlier attempts to characterize the behavior of simulation algorithms in our explicit use of Bayesian statistics: We update a probability distribution over target probabilities of interest with information from randomized trials. For real ffl; ffi ! 1 and for a probabilistic inference Pr[xje], the output of an inference approximation algorithm is an (ffl; ffi)estimate of Pr[xje] if with probability at least 1 \Gamma ffi the output is within relative error ffl of Pr[xje]. We construct a stopping rule for the number of simulations required by logic sampling, randomized approximation schemes, and likelihood weighting to provide (ffl; ffi)estimates of Pr[xje]. With probability 1 \Gamma ffi, the stopping rule is optimal in the sense that the algorithm performs the minimum number of required simulations. We prove that our stopping rules are insensitive to the prior probability distribution on Pr[xje].
Possible biases induced by MCMC convergence diagnostics
, 1997
"... This paper is organised as follows. In Section 2, we present an oversimplified version of a convergence diagnostic, and study analytically its performance on certain simple Markov chains. We restrict ourselves primarily to chains which in fact produce i.i.d. samples from ..."
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Cited by 16 (2 self)
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This paper is organised as follows. In Section 2, we present an oversimplified version of a convergence diagnostic, and study analytically its performance on certain simple Markov chains. We restrict ourselves primarily to chains which in fact produce i.i.d. samples from