On the Poisson equation for Metropolis–Hastings chains

Abstract

This paper defines an approximation scheme for a solution of the Poisson equation of a geometrically ergodic Metropolis–Hastings chain $\Phi$. The scheme is based on the idea of weak approximation and gives 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 results show that the sequence of the asymptotic variances (in the CLTs for the control-variate estimators) converges to zero and give a rate of this convergence. Numerical examples in the case of a double-well potential are discussed.