Introduction to MCMC
Although Markov chain Monte Carlo (MCMC) techniques have been available since Metropolis and Ulam (1949), which is almost as long as the invention of computational Monte Carlo techniques in the 1940s by the Los Alamos physicists working on the atomic bomb, they have only been popular in mainstream statistics since the pioneering paper of Gelfand and Smith (1990) and the subsequent papers in the early 1990s. Gelfand and Smith (1990) introduced Gibbs sampling to the statistics community. It is no coincidence that the BUGS project started in 1989 in Cambridge, UK, and was led by David Spiegelhalter, who had been a PhD student of Adrian Smith's at Oxford. Both share a passion for Bayesian statistics. Recent accounts of MCMC techniques can be found in the book by Gamerman and Lopes (2006) or in Robert and Casella (2011).
Hastings (1970) generalized the Metropolis algorithm but the idea had remained unused in the statistics literature. It was soon realized that Metropolis–Hastings could be used within Gibbs for those situations where it was difficult to implement so-called pure Gibbs. With a clear connection between the expectation–maximization (EM) algorithm, for obtaining modal values of likelihoods or posteriors where there are missing values or latent values, and Gibbs sampling, MCMC approaches were developed for models where there are latent ...