Control Variates for the Metropolis-Hastings Algorithm

We propose new control variates for variance reduction in estimation of mean values using the Metropolis-Hastings algorithm. Traditionally, states that are rejected in the Metropolis-Hastings algorithm are simply ignored, which intuitively seems to be a waste of information. We present a setting for construction of zero mean control variates for general target and proposal distributions and develop ideas for the standard Metropolis-Hastings and reversible jump algorithms. We give results for three simulation examples. We get best results for variates that are functions of the current state <b>x</b> and the proposal <b>y</b>, but we also consider variates that in addition are functions of the Metropolis-Hastings acceptance/rejection decision. The variance reduction achieved varies depending on the target distribution and proposal mechanisms used. In simulation experiments, we typically achieve relative variance reductions between 15% and 35%. Copyright (c) Board of the Foundation of the Scandinavian Journal of Statistics 2008.