Annals of Statistics

Markov Chains for Exploring Posterior Distributions

Luke Tierney

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Several Markov chain methods are available for sampling from a posterior distribution. Two important examples are the Gibbs sampler and the Metropolis algorithm. In addition, several strategies are available for constructing hybrid algorithms. This paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues. On the theoretical side, results from the theory of general state space Markov chains can be used to obtain convergence rates, laws of large numbers and central limit theorems for estimates obtained from Markov chain methods. These theoretical results can be used to guide the construction of more efficient algorithms. For the practical use of Markov chain methods, standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation.

Article information

Ann. Statist., Volume 22, Number 4 (1994), 1701-1728.

First available in Project Euclid: 11 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 65C05: Monte Carlo methods

62-04 Monte Carlo Metropolis-Hastings algorithm Gibbs sampler variance reduction


Tierney, Luke. Markov Chains for Exploring Posterior Distributions. Ann. Statist. 22 (1994), no. 4, 1701--1728. doi:10.1214/aos/1176325750.

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