Practical Markov Chain Monte Carlo

Markov chain Monte Carlo using the Metropolis-Hastings algorithm is a general method for the simulation of stochastic processes having probability densities known up to a constant of proportionality. Despite recent advances in its theory, the practice has remained controversial. This article makes the case for basing all inference on one long run of the Markov chain and estimating the Monte Carlo error by standard nonparametric methods well-known in the time-series and operations research literature. In passing it touches on the Kipnis-Varadhan central limit theorem for reversible Markov chains, on some new variance estimators, on judging the relative efficiency of competing Monte Carlo schemes, on methods for constructing more rapidly mixing Markov chains and on diagnostics for Markov chain Monte Carlo.