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 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.

Presentation streamlined, new short proof of Proposition 3.2 in the reversible case with other arguments essentially unchanged, 25 pages, no figures, to appear in Bernoulli