Output Analysis for Markov Chain Monte Carlo

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Markov chain Monte Carlo (MCMC) has several disadvantages when compared to classical Monte Carlo methods. In particular, an important issue that every practitioner faces when implementing MCMC is when to stop the computation. Typically, a mixture of experience and ad hoc methods is employed to make this decision. Thus one is forced to wonder about the quality of the inference. The investigator studies sequential fixed-width methods that allow construction of an interval estimator for the quantity of interest. The interval estimator describes the confidence in the point estimate. The investigator uses this to study the development of valid stopping rules when the MCMC computation is aimed at estimating general quantities of the target distribution. These methods require the Markov chain to converge at a geometric rate which in turn implies there is a limiting distribution of the point estimate in the settings of interest. Thus the investigator studies the convergence rates of Markov chains encountered in two broad classes of Bayesian models. MCMC methods have become a standard technique in the toolbox of applied statisticians (and many scientists in other disciplines). Indeed, it is not much of an overstatement to say that it has revolutionized applied statistics, especially that of the Bayesian variety. Unfortunately, MCMC methods are not always used carefully leading to dubious claims in the literature. In particular, there has been little effort to include measures of uncertainty in inferential conclusions. Rigorously addressing the issue of stopping rules in terms of these measures of uncertainty enhances infrastructure for research and education by providing statisticians and other scientists valid techniques for using MCMC to make inference in their research setting.