Markov Chain Monte Carlo A Contribution to the Encyclopedia of Environmetrics

Realistic statistical models often give rise to probability distributions that are computationally difficult to use for inference. Fortunately, we now have a collection of algorithms, known as Markov chain Monte Carlo (MCMC), that has brought many of these models within our computational reach. In turn, this has lead to a staggering amount of both theoretical and applied work on MCMC. Thus we do not propose a complete overview of MCMC in this article. Actually, we only hope to get the reader started in the right direction. To this end, we spend some time indicating when it is appropriate to consider using MCMC. Then we introduce the fundamental algorithms and address some general implementation issues. An obvious omission from this paper is an introduction to the Markov chain theory underpinning MCMC algorithms. Instead we refer the interested reader to Besag, Green, Higdon & Mengersen (1995), Robert & Casella (1999), and Roberts & Rosenthal (1998). However, throughout we will use the phrases “MCMC algorithm” and “Markov chain” interchangeably.