Multivariate output analysis for Markov chain Monte Carlo

• 成果类型: Article
• 署名作者: Vats, Dootika; Flegal, James M.; Jones, Galin L.
• 署名单位: University of Warwick; University of California System; University of California Riverside; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asz002
• 发表日期: 2019
• 页码: 321337
• 关键词: spectral variance estimators; sure invariance-principles; geometric ergodicity; strong consistency; gibbs samplers; ASYMPTOTIC VARIANCE; time-series; CONVERGENCE; approximation; univariate

摘要

Markov chain Monte Carlo produces a correlated sample which may be used 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. The multivariate nature of this Monte Carlo error has been largely ignored in the literature. We present a multivariate framework for terminating a simulation in Markov chain Monte Carlo. We define a multivariate effective sample size, the estimation of which requires strongly consistent estimators of the covariance matrix in the Markov chain central limit theorem, 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. This lower bound does not depend on the underlying stochastic process and can be calculated a priori. This result is obtained by drawing a connection between terminating simulation via effective sample size and terminating simulation using a relative standard deviation fixed-volume sequential stopping rule, which we demonstrate is an asymptotically valid procedure. The finite-sample properties of the proposed method are demonstrated in a variety of examples.