Markov chain Monte Carlo in statistical mechanics: The problem of accuracy

The appearance of the article by N. Metropolis. A.W. Rosenbluth, M.N. Rosenbluth, A. H. Teller, and E. Teller marked the birth of the Monte Carlo method for the study of statistical-mechanical systems and of a specific form of "importance sampling"-namely, Markov chain Monte Carlo. After nearly 40 years of statistical usage, this technique has had a profound impact on statistical theory, on both Bayesian and classical statistics. Markov chain Monte Carlo is used essentially to estimate integrals in high dimensions. This article addresses the accuracy of such estimation. Through computer experiments performed on the two-dimensional Ising model, we compare the most common method for error estimates in statistical mechanics. It appears that the moving-block bootstrap outperforms other methods based on subseries values when the number of observations is relatively small and the time correlation between successive configurations decays slowly. Moreover, the moving-block bootstrap enables estimates of the standard error to be made not only for the averages of directly obtained data but also for estimates derived from sophisticated numerical procedures.