Variance Reduction for Metropolis-Hastings Samplers

TL;DR

This paper presents a versatile variance reduction technique for Metropolis-Hastings algorithms that leverages control variates derived from approximate solutions to the Poisson equation, significantly improving estimator efficiency with minimal additional computation.

Contribution

It introduces a novel post-processing variance reduction framework using Gaussian approximations and Poisson equation solutions for Metropolis algorithms, enhancing estimator accuracy.

Findings

Significant variance reduction demonstrated in Gaussian target scenarios.

Effective variance reduction observed in Bayesian logistic regression.

Method incurs negligible additional computational cost.

Abstract

We introduce a general framework that constructs estimators with reduced variance for random walk Metropolis and Metropolis-adjusted Langevin algorithms. The resulting estimators require negligible computational cost and are derived in a post-process manner utilising all proposal values of the Metropolis algorithms. Variance reduction is achieved by producing control variates through the approximate solution of the Poisson equation associated with the target density of the Markov chain. The proposed method is based on approximating the target density with a Gaussian and then utilising accurate solutions of the Poisson equation for the Gaussian case. This leads to an estimator that uses two key elements: (i) a control variate from the Poisson equation that contains an intractable expectation under the proposal distribution, (ii) a second control variate to reduce the variance of a Monte Carlo estimate of this latter intractable expectation. Simulated data examples are used to illustrate the impressive variance reduction achieved in the Gaussian target case and the corresponding effect when target Gaussianity assumption is violated. Real data examples on Bayesian logistic regression and stochastic volatility models verify that considerable variance reduction is achieved with negligible extra computational cost.