Output Analysis for Markov Chain Monte Carlo

Abstract

Markov chain Monte Carlo (MCMC) is a sampling method used to estimate expectations with respect to a target distribution. An important question is when should sampling stop so that we have good estimates of these expectations? The key to answering this question lies in assessing the Monte Carlo error through a multivariate Markov chain central limit theorem (CLT). The multivariate nature of this Monte Carlo error largely has been ignored in the MCMC literature. This dissertation discusses the drawbacks of the current univariate methods of terminating simulation and introduces a multivariate framework for terminating simulation. Theoretical properties of the procedures are established. A multivariate effective sample size is defined and estimated using strongly consistent estimators of the covariance matrix in the Markov chain CLT, a property that is shown for the multivariate batch means estimator and the multivariate spectral variance estimator. A critical aspect of this procedure is that a lower bound on the number of effective samples required for a pre-specified level of precision can be determined a priori. This lower bound depends on the problem only in the dimension of the expectation being estimated, and not on the underlying stochastic process. This result is obtained by drawing a connection between terminating the simulation via effective sample size and terminating it using a relative standard deviation fixed-volume sequential stopping rule. The finite sample properties of the proposed methods are demonstrated in a variety of examples. The proposed method requires the existence of a Markov chain CLT, establishing which requires bounding the rate of convergence of the Markov chains. We establish a geometric rate of convergence for a class of Bayesian penalized regression models. We also present examples showing how it may be easier to establish rates of convergence for linchpin variable samplers than for its competitors.