On the Poisson equation for Metropolis-Hastings chains

Aleksandar Mijatovic, Jure Vogrinc

Published 2015-11-23Version 1

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. We apply the algorithm to geometrically and non-geometrically ergodic chains and present numerical evidence for a significant variance reduction in both cases.