Markov Chain Monte Carlo Methods

Abstract

Markov chain Monte Carlo methods are used to sample from complicated multivariate distributions that are not computable in practice and from which direct sampling (Monte Carlo) is not feasible. In the field of genetics, they are used in pedigree and linkage analysis to obtain the posterior probabilities of individuals possessing specific genes.

Keywords: Gibbs sampler; Metropolis–Hastings; pedigree analysis; linkage analysis; ergodic; blocking; blocked

References

Besag J (2000) Markov Chain Monte Carlo for Statistical Inference. Seattle, WA: University of Washington.

Gelfand AE and Smith AFM (1990) Sampling‐based approaches to calculating marginal densities. Journal of the American Statistical Association 85(410): 398–409.

Geyer CJ (1992) Practical Markov chain Monte Carlo. Statistical Science 7(4): 473–511.

Geyer CJ and Thompson EA (1995) Annealing Markov chain Monte Carlo with applications to ancestral inference. Journal of the American Statistical Association 90(431): 909–920.

Gilks WR, Richardson S and Spiegelhalter DJ (eds.) (1996) Markov Chain Monte Carlo in Practice. London, UK: Chapman & Hall.

Sheehan N and Thomas A (1993) On the irreducibility of a Markov chain defined on a space of genotype configurations by a sampling scheme. Biometrics 49: 163–175.

Skaanning JC (1997) Blocking Gibbs Sampling for Inference in Large and Complex Bayesian Networks with Applications in Genetics. PhD thesis, May 1997/ Technical Report R‐97‐5005, Department of Computer Science, Aalborg University, Denmark.

Skaanning JC and Kong A (1999) Blocking Gibbs sampling for linkage analysis in large pedigrees with many loops. American Journal of Human Genetics 65(3): 885–902.

Skaanning JC and Sheehan N (1998) Problems with the determination of noncommunicating classes for MCMC applications in pedigree analysis. Biometrics 54(2): 416–425.

Smith AFM and Roberts GO (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, Series B 55(1): 5–23.

Thomas A, Gutin A, Abkevich V and Bansal A (2000) Multilocus linkage analysis by blocked Gibbs sampling. Statistics and Computing 10: 259–269.

Tierney L (1994) Markov chains for exploring posterior distributions (with discussion). Annals of Statistics 22: 1701–1762.

Further Reading

Besag J, Green P, Higdon D and Mengersen K (1995) Bayesian computation and stochastic systems (with discussion). Statistical Science 10(1): 3–66.

Gamerman P (1997) Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. London, UK: Chapman & Hall.

Green PJ (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82: 711–732.

Kass RE, Carlin BP, Gelman A and Neal RM (1998) Markov chain Monte Carlo in practice: a roundtable discussion. American Statistician 52: 93–100.

Neal RM (1993) Probabilistic Inference Using Markov Chain Monte Carlo Methods. Technical Report CRG‐TR‐93‐1, Department of Computer Science, University of Toronto.

Robert CP and Casella G (1999) Monte Carlo Statistical Methods. New York, NY: Springer‐Verlag.

Web Links

Markov Chain Monte Carlo for Statistical Inference. Citeseer entry for the 2000 reference paper of Julian Besag. On this page you will find links to many papers on Markov chains http://citeseer.nj.nec.com/besag00markov.html

Radford Neal's Research: Markov Chain Monte Carlo. Overview of Radford Neal's research on Markov chains, including links to his reference papers http://www.cs.toronto.edu/∼radford/res‐mcmc.html

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How to Cite close
Skaanning, Claus(Jul 2006) Markov Chain Monte Carlo Methods. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1038/npg.els.0005437]