Two important questions that must be answered whenever a Markov chain Monte Carlo (MCMC) algorithm is used are (Q1) What is an appropriate burn-in? and (Q2) How long should the sampling continue after burn-in?Developing rigorous answers to these questions presently requires a detailed study of the convergence properties of the underlying Markov chain. Consequently, in most practical applications of MCMC, exact answers to (Q1)and (Q2) are not sought. The goal of this paper is to demystify the analysis that leads to honest answers to (Q1) and (Q2). The authors hope that this article will serve as a bridge between those developing Markov chain theory and practitioners using MCMC to solve practical problems.

The ability to address (Q1) and (Q2) formally comes from establishing a drift condition and an associated minorization condition, which together imply that the underlying Markov chain is geometrically ergodic. In this article, we explain exactly what drift and minorization are as well as how and why these conditions can be used to form rigorous answers to (Q1) and (Q2). The basic ideas are as follows. The results of Rosenthal (1995) and Roberts and Tweedie (1999) allow one to use drift and minorization conditions to construct a formula giving an analytic upper bound on the distance to stationarity. A rigorous answer to (Q1) can be calculated using this formula. The desired characteristics of the target distribution are typically estimated using ergodic averages. Geometric ergodicity of the underlying Markov chain implies that there are central limit theorems available for ergodic averages (Chan and Geyer 1994). The regenerative simulation technique (Mykland, Tierney and Yu, 1995; Robert, 1995) can be used to get a consistent estimate of the variance of the asymptotic normal distribution. Hence, an asymptotic standard error can be calculated, which provides an answer to (Q2) in the sense that an appropriate time to stop sampling can be determined. The methods are illustrated using a Gibbs sampler for a Bayesian version of the one-way random effects model and a data set concerning styrene exposure.

The Metropolis–Hastings algorithm transforms a given stochastic matrix into a reversible stochastic matrix with a prescribed stationary distribution. We show that this transformation gives the minimum distance solution in an $L^1$ metric.

An overview of orderings defined on the space of Markov chains having a prespecified unique stationary distribution is given. The intuition gained by studying these orderings is used to improve existing Markov chain Monte Carlo algorithms.

We review and extend results related to optimal scaling of Metropolis–Hastings algorithms. We present various theoretical results for the high-dimensional limit. We also present simulation studies which confirm the theoretical results in finite-dimensional contexts.

Richard E. Barlow was one of the two founders of modern reliability theory (Frank Proschan was the other). Two of their books [Barlow and Proschan, 1965, 1975] have been influential in shaping this field. Barlow was born in 1931 in Galesburg, Illinois. He received his bachelor’s degree in mathematics from Knox College in 1953, his master’s in mathematics from the University of Oregon in 1955 and his Ph.D. from Stanford University in 1960. He worked at the Institute of Defense Analysis (1960 –1961) and General Telephone (1961 –1963) before joining the University of California, Berkeley, in the Department of Industrial Engineering and Operations Research in 1963. He was at Berkeley until his retirement in 1999 and is now Professor Emeritus. He has visited the Boeing Scientific Laboratories (1966) and Florida State University (1975 –1976) as well as many other places. He has been an associate editor of most of the major statistics and operations research journals. He is a Fellow of both the American Statistical Association and the Institute of Mathematical Statistics.

In honor of his many accomplishments, Richard E.Barlow was awarded the Von Neumann Prize of the Operations Research Society in 1991 jointly with Frank Proschan.