Multivariate Output Analysis for Markov Chain Monte Carlo

• Activity Number: 307 - Bayesian Computational Advances for Complex and Large-Scale Data
• Type: Invited
• Date/Time: Tuesday, August 1, 2017 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Computing
• Author(s): Galin Jones* and Dootika Vats and James Flegal
• Companies: University of Minnesota and University of Minnesota and University of California, Riverside
• Keywords: Markov chain ; Monte Carlo ; Standard Error ; Simulation ; Termination ; Effective Sample Size

Abstract

Markov chain Monte Carlo (MCMC) produces a correlated sample for estimating expectations with respect to a target distribution. A fundamental question is when should sampling stop so that we have good estimates of the desired quantities? 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. We present a multivariate framework for terminating simulation in MCMC. We define a multivariate effective sample size, estimating which requires strongly consistent estimators of the asymptotic covariance matrix in the Markov chain CLT; a property we show for the multivariate batch means estimator. We then provide a lower bound on the number of minimum effective samples required for a desired level of precision.