Implementing MCMC: Multivariate estimation with confidence

TL;DR

This paper reviews methods for estimating the asymptotic covariance in MCMC, focusing on practical, efficient techniques to improve the accuracy of simulation results, especially under positive correlation.

Contribution

It provides a comprehensive overview of batching, spectral, and initial sequence covariance estimation methods with practical recommendations for modern MCMC.

Findings

Spectral and batching methods are effective for covariance estimation.

Positive correlation in MCMC leads to negatively biased estimates.

Efficient methods can improve the reliability of MCMC output.

Abstract

This paper addresses the key challenge of estimating the asymptotic covariance associated with the Markov chain central limit theorem, which is essential for visualizing and terminating Markov Chain Monte Carlo (MCMC) simulations. We focus on summarizing batching, spectral, and initial sequence covariance estimation techniques. We emphasize practical recommendations for modern MCMC simulations, where positive correlation is common and leads to negatively biased covariance estimates. Our discussion is centered on computationally efficient methods that remain viable even when the number of iterations is large, offering insights into improving the reliability and accuracy of MCMC output in such scenarios.