Multivariate moment least-squares estimators for reversible Markov chains

TL;DR

This paper introduces new multivariate autocovariance and asymptotic variance estimators for reversible Markov chains, enhancing uncertainty quantification in MCMC by capturing cross-correlations among multiple components.

Contribution

It extends the univariate moment least squares estimator to multivariate functions, providing consistent estimators for autocovariance sequences and variance matrices in reversible Markov chains.

Findings

Proposed estimators are strongly consistent.

Empirical comparisons show improved performance over existing methods.

Effective in both simulated and real-data scenarios.

Abstract

Markov chain Monte Carlo (MCMC) is a commonly used method for approximating expectations with respect to probability distributions. Uncertainty assessment for MCMC estimators is essential in practical applications. Moreover, for multivariate functions of a Markov chain, it is important to estimate not only the auto-correlation for each component but also to estimate cross-correlations, in order to better assess sample quality, improve estimates of effective sample size, and use more effective stopping rules. Berg and Song [2022] introduced the moment least squares (momentLS) estimator, a shape-constrained estimator for the autocovariance sequence from a reversible Markov chain, for univariate functions of the Markov chain. Based on this sequence estimator, they proposed an estimator of the asymptotic variance of the sample mean from MCMC samples. In this study, we propose novel autocovariance sequence and asymptotic variance estimators for Markov chain functions with multiple components, based on the univariate momentLS estimators from Berg and Song [2022]. We demonstrate strong consistency of the proposed auto(cross)-covariance sequence and asymptotic variance matrix estimators. We conduct empirical comparisons of our method with other state-of-the-art approaches on simulated and real-data examples, using popular samplers including the random-walk Metropolis sampler and the No-U-Turn sampler from STAN.