Control Variates for the Metropolis–Hastings Algorithm

Abstract.  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 x and the proposal y , 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%.