Statistical Science

Practical Markov Chain Monte Carlo

Charles J. Geyer

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Abstract

Markov chain Monte Carlo using the Metropolis-Hastings algorithm is a general method for the simulation of stochastic processes having probability densities known up to a constant of proportionality. Despite recent advances in its theory, the practice has remained controversial. This article makes the case for basing all inference on one long run of the Markov chain and estimating the Monte Carlo error by standard nonparametric methods well-known in the time-series and operations research literature. In passing it touches on the Kipnis-Varadhan central limit theorem for reversible Markov chains, on some new variance estimators, on judging the relative efficiency of competing Monte Carlo schemes, on methods for constructing more rapidly mixing Markov chains and on diagnostics for Markov chain Monte Carlo.

Article information

Source
Statist. Sci. Volume 7, Number 4 (1992), 473-483.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.ss/1177011137

Digital Object Identifier
doi:10.1214/ss/1177011137

Zentralblatt MATH identifier

JSTOR
links.jstor.org

Keywords
Markov chain Monte Carlo Metropolis-Hastings algorithm Gibbs sampler central limit theorem variance estimation

Citation

Geyer, Charles J. Practical Markov Chain Monte Carlo. Statist. Sci. 7 (1992), no. 4, 473--483. doi:10.1214/ss/1177011137. http://projecteuclid.org/euclid.ss/1177011137.


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See also

  • See Comment: Lu Cui, Martin A. Tanner, Debajyoti Sinha, W. J. Hall. [Practical Markov Chain Monte Carlo]: Comment: Monitoring Convergence of the Gibbs Sampler: Further Experience with the Gibbs Stopper. Statist. Sci., Volume 7, Number 4 (1992), 483--486.
  • See Comment: Alan E. Gelfand. [Practical Markov Chain Monte Carlo]: Comment. Statist. Sci., Volume 7, Number 4 (1992), 486--487.
  • See Comment: Neal Madras. [Practical Markov Chain Monte Carlo]: Comment. Statist. Sci., Volume 7, Number 4 (1992), 488--489.
  • See Comment: Nicholas G. Polson. [Practical Markov Chain Monte Carlo]: Comment. Statist. Sci., Volume 7, Number 4 (1992), 490--491.
  • See Comment: Amy Racine-Poon. [Practical Markov Chain Monte Carlo]: Comment. Statist. Sci., Volume 7, Number 4 (1992), 492--493.
  • See Comment: Adrian E. Raftery, Steven M. Lewis. [Practical Markov Chain Monte Carlo]: Comment: One Long Run with Diagnostics: Implementation Strategies for Markov Chain Monte Carlo. Statist. Sci., Volume 7, Number 4 (1992), 493--497.
  • See Comment: Jeffrey S. Rosenthal. [Practical Markov Chain Monte Carlo]: Comment. Statist. Sci., Volume 7, Number 4 (1992), 498--498.
  • See Comment: Bruce Schmeiser. [Practical Markov Chain Monte Carlo]: Comment. Statist. Sci., Volume 7, Number 4 (1992), 498--499.
  • See Comment: Luke Tierney. [Practical Markov Chain Monte Carlo]: Comment. Statist. Sci., Volume 7, Number 4 (1992), 499--501.
  • See Comment: Charles J. Geyer. [Practical Markov Chain Monte Carlo]: Rejoinder. Statist. Sci., Volume 7, Number 4 (1992), 502--503.
  • See Comment: Andrew Gelman, Donald B. Rubin. [Practical Markov Chain Monte Carlo]: Rejoinder: Replication without Contrition. Statist. Sci., Volume 7, Number 4 (1992), 503--511.