Markov chain Monte Carlo method and its application

Abstract

The Markov chain Monte Carlo (MCMC) method, as a computer-intensive statistical tool, has enjoyed an enormous upsurge in interest over the last few years. This paper provides a simple, comprehensive and tutorial review of some of the most common areas of research in this ®eld. We begin by discussing how MCMC algorithms can be constructed from standard buildingblocks to produce Markov chains with the desired stationary distribution. We also motivate and discuss more complex ideas that have been proposed in the literature, such as continuous time and dimension jumping methods. We discuss some implementational issues associated with MCMC methods. We take a look at the arguments for and against multiple replications, consider how long chains should be run for and how to determine suitable starting points. We also take a look at graphical models and how graphical approaches can be used to simplify MCMC implementation. Finally, we present a couple of examples, which we use as case-studies to highlight some of the points made earlier in the text. In particular, we use a simple changepoint model to illustrate how to tackle a typical Bayesian modelling problem via the MCMC method, before using mixture model problems to provide illustrations of good sampler output and of the implementation of a reversible jump MCMC algorithm.

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“…This has been made possible, at least approximately, by the upsurge of Markov Chain Monte Carlo methods (MCMC, for a review see e.g. Brooks (1998)). …”

Spatial statistical methods in health

“…In the MCMC approach the full hyperprior framework is used, but rather than attempt to determine P( , ␥|y) by direct mathematical analysis, instead observations are indirectly simulated from this posterior using MCMC methods (e.g. see Brooks, 1998;Gilks et al, 1996). The desired parameter estimates ˆa re then calculated from relevant sample statistics of the simulated values from P( , ␥|y).…”

Variable selection in censored kernel regression

“…The estimating equations are solved through iterative method with Gehan (1965)-type estimate as the initial value. Ghosh and Ghosal (2006) proposed the estimation procedure based on a nonparametric Bayesian approach which uses a Dirichlet prior for the mixture of Weibull distribution in the censored regression model, where Markov Chain Monte Carlo method (Brooks, 1998) is used to obtain the marginal posterior distribution of regression parameters. Shim et al (2011) proposed a semiparametric LS-SVM for the censored data using weights which Koul et al (1981) proposed.…”