Bayesian Computation Via Markov Chain Monte Carlo

A search for Markov chain Monte Carlo (or MCMC) articles on Google Scholar yields over 100,000 hits, and a general web search on Google yields 1.7 million hits. These results stem largely from the ubiquitous use of these algorithms in modern computational statistics, as we shall now describe. MCMC algorithms are used to solve problems in many scientific fields, including physics (where many MCMC algorithms originated) and chemistry and computer science. However, the widespread popularity of MCMC samplers is largely due to their impact on solving statistical computation problems related to Bayesian inference. Specifically, suppose we are given an independent and identically distributed (henceforth iid) sample {x1, . . . , xn} from a parametric sampling density f(x|θ), where x ∈ X ⊂ R and θ ∈ Θ ⊂ R. Suppose we also have some prior density p(θ). Then the Bayesian paradigm prescribes that all aspects of inference should be based on the