Notes on Using Control Variates for Estimation with Reversible MCMC Samplers

TL;DR

This paper introduces a rigorous methodology for constructing and using control variates in reversible MCMC samplers, significantly reducing variance in Bayesian inference applications.

Contribution

It provides a novel, theoretically justified approach for optimal control variate coefficients and adaptive estimators in reversible MCMC methods.

Findings

Variance reduction can be substantial in Bayesian inference tasks.

Methodology is rigorously justified with asymptotic analysis.

Numerical examples demonstrate practical effectiveness.

Abstract

A general methodology is presented for the construction and effective use of control variates for reversible MCMC samplers. The values of the coefficients of the optimal linear combination of the control variates are computed, and adaptive, consistent MCMC estimators are derived for these optimal coefficients. All methodological and asymptotic arguments are rigorously justified. Numerous MCMC simulation examples from Bayesian inference applications demonstrate that the resulting variance reduction can be quite dramatic.