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Markov Chain Monte Carlo Technology

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  • Chib, Siddhartha

Abstract

In the past fifteen years computational statistics has been enriched by a powerful, somewhat abstract method of generating variates from a target probability distribution that is based on Markov chains whose stationary distribution is the probability distribution of interest. This class of methods, popularly referred to as Markov chain Monte Carlo methods, or simply MCMC methods, have been influential in the modern practice of Bayesian statistics where these methods are used to summarize the posterior distributions that arise in the context of the Bayesian prior-posterior analysis (Tanner and Wong, 1987; Gelfand and Smith, 1990; Smith and Roberts, 1993; Tierney, 1994; Besaget al., 1995; Chib and Greenberg, 1995, 1996; Gilks et al., 1996; Tanner, 1996; Gammerman, 1997; Robert and Casella, 1999; Carlin and Louis, 2000; Chen et al., 2000; Chib, 2001; Congdon, 2001; Liu, 2001; Robert, 2001; Gelman at al, 2003). MCMC methods have proved useful in practically all aspects of Bayesian inference, for example, in the context of prediction problems and in the computation of quantities, such as the marginal likelihood, that are used for comparing competing Bayesian models.

Suggested Citation

  • Chib, Siddhartha, 2004. "Markov Chain Monte Carlo Technology," Papers 2004,22, Humboldt University of Berlin, Center for Applied Statistics and Economics (CASE).
    • Handle: RePEc:zbw:caseps:200422
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    References listed on IDEAS

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    1. Geweke, John, 1989. "Bayesian Inference in Econometric Models Using Monte Carlo Integration," Econometrica, Econometric Society, vol. 57(6), pages 1317-1339, November.
    2. Chib, Siddhartha & Greenberg, Edward, 1994. "Bayes inference in regression models with ARMA (p, q) errors," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 183-206.
    3. Chib, Siddhartha, 2001. "Markov chain Monte Carlo methods: computation and inference," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 5, chapter 57, pages 3569-3649, Elsevier.
    4. Antonietta Mira & Luke Tierney, 2002. "Efficiency and Convergence Properties of Slice Samplers," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 29(1), pages 1-12, March.
    5. P. Damlen & J. Wakefield & S. Walker, 1999. "Gibbs sampling for Bayesian non‐conjugate and hierarchical models by using auxiliary variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(2), pages 331-344, April.
    6. Chib, Siddhartha & Greenberg, Edward, 1996. "Markov Chain Monte Carlo Simulation Methods in Econometrics," Econometric Theory, Cambridge University Press, vol. 12(3), pages 409-431, August.
    7. Gareth O. Roberts & Jeffrey S. Rosenthal, 1999. "Convergence of Slice Sampler Markov Chains," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 643-660.
    8. Chib, Siddhartha & Greenberg, Edward & Winkelmann, Rainer, 1998. "Posterior simulation and Bayes factors in panel count data models," Journal of Econometrics, Elsevier, vol. 86(1), pages 33-54, June.
    9. Chib S. & Jeliazkov I., 2001. "Marginal Likelihood From the Metropolis-Hastings Output," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 270-281, March.
    10. G. O. Roberts & S. K. Sahu, 1997. "Updating Schemes, Correlation Structure, Blocking and Parameterization for the Gibbs Sampler," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(2), pages 291-317.
    11. Roberts, G. O. & Smith, A. F. M., 1994. "Simple conditions for the convergence of the Gibbs sampler and Metropolis-Hastings algorithms," Stochastic Processes and their Applications, Elsevier, vol. 49(2), pages 207-216, February.
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