On the Poisson equation for Metropolis-Hastings chains

TL;DR

This paper introduces an approximation scheme for solving the Poisson equation in geometrically ergodic Metropolis-Hastings chains, leading to improved control variates that reduce asymptotic variance in ergodic averages.

Contribution

It proposes a novel approximation method for the Poisson equation that enhances control variate techniques in Markov chain Monte Carlo methods.

Findings

Asymptotic variances converge to zero with a quantifiable rate.

Numerical examples demonstrate effectiveness in double-well potential scenarios.

Abstract

This paper defines an approximation scheme for a solution of the Poisson equation of a geometrically ergodic Metropolis-Hastings chain $\Phi$. The approximations give rise to a natural sequence of control variates for the ergodic average $S_k(F)=(1/k)\sum_{i=1}^{k} F(\Phi_i)$, where $F$ is the force function in the Poisson equation. The main result of the paper shows that the sequence of the asymptotic variances (in the CLTs for the control-variate estimators) converges to zero and gives a rate of this convergence. Numerical examples in the case of a double-well potential are discussed.