Markov Chain Monte Carlo Sampling
Recently, Monte Carlo (MC) based sampling methods for evaluating high-dimensional posterior integrals have been rapidly developing. Those sampling methods include MC importance sampling (Hammersley and Handscomb 1964; Ripley 1987; Geweke 1989; Wolpert 1991), Gibbs sampling (Geman and Geman 1984; Gelfand and Smith 1990), Hit-and-Run sampling (Smith 1984; Bélislc, Romeijn, and Smith 1993; Chen 1993; Chen and Schmeiser 1993 and 1996), Metropolis-Hastings sampling (Metropolis et al. 1953; Hastings 1970; Green 1995), and hybrid methods (e.g., Müller 1991; Tierney 1994; Berger and Chen 1993). A general discussion of the Gibbs sampler and other Markov chain Monte Carlo (MCMC) methods is given in the Journal of the Royal Statistical Society, Series B (1993), and an excellent roundtable discussion on the practical use of MCMC can be found in Kass et al. (1998). Other discussions or instances of the use of MCMC sampling can be found in Tanner and Wong (1987), Tanner (1996), Geyer (1992), Gelman and Rubin (1992), Gelfand, Smith, and Lee (1992), Gilks and Wild (1992), and many others. Further development of state-of-the-arts MCMC sampling techniques include the accelerated MCMC sampling of Liu and Sabatti (1998, 1999), Liu (1998), and Liu and Wu (1997), and the exact MCMC sampling of Green and Murdoch (1999). Comprehensive accounts of MCMC methods and their applications may also be found in Meyn and Tweedie (1993), Tanner (1996), Gilks, Richardson, and Spiegel-halter (1996), Robert and Casella (1999), and Gelfand and Smith (2000). The purpose of this chapter is to give a brief overview of several commonly used MCMC sampling algorithms as well as to present selectively several newly developed computational tools for MCMC sampling.
KeywordsPosterior Distribution Markov Chain Monte Carlo Monte Carlo Markov Chain Monte Carlo Algorithm Markov Chain Monte Carlo Sampling
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