Globally-centered autocovariances in MCMC

TL;DR

This paper introduces a globally-centered autocovariance estimator for MCMC that reduces bias and improves accuracy in slow-mixing chains, enhancing analysis tools like ACF plots and effective sample size estimates.

Contribution

It proposes a novel G-ACvF estimator with theoretical guarantees and empirical improvements over existing methods for MCMC autocovariance estimation.

Findings

G-ACvF has smaller bias than existing estimators.

The estimator is strongly consistent under weak conditions.

Performance improvements are demonstrated through multiple examples.

Abstract

Autocovariances are a fundamental quantity of interest in Markov chain Monte Carlo (MCMC) simulations with autocorrelation function (ACF) plots being an integral visualization tool for performance assessment. Unfortunately, for slow-mixing Markov chains, the empirical autocovariance can highly underestimate the truth. For multiple-chain MCMC sampling, we propose a globally-centered estimator of the autocovariance function (G-ACvF) that exhibits significant theoretical and empirical improvements. We show that the bias of the G-ACvF estimator is smaller than the bias of the current state-of-the-art. The impact of this improved estimator is evident in three critical output analysis applications: (1) ACF plots, (2) estimates of the Monte Carlo asymptotic covariance matrix, and (3) estimates of the effective sample size. Under weak conditions, we establish strong consistency of our improved asymptotic covariance estimator, and obtain its large-sample bias and variance. The performance of the new estimators is demonstrated through various examples.