Abstract

Interest is in evaluating, by Markov chain Monte Carlo (MCMC) simulation, the expected value of a function with respect to a, possibly unnormalized, probability distribution. A general purpose variance reduction technique for the MCMC estimator, based on the zero-variance principle introduced in the physics literature, is proposed. The main idea is to construct control variates based on the score function. Conditions for asymptotic unbiasedness of the zero-variance estimator are derived. A central limit theorem is also proved under regularity conditions. The potential of the zero-variance strategy is illustrated with real applications to probit, logit and GARCH Bayesian models. The Zero-Variance principle is efficiently combined with Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms without exceeding the computational requirements since its main ingredient (namely the score function) is exploited twice: once to guide the Markov chain towards relevant portion of the state space via a clever proposal, that exploits the geometry of the target and achieves convergence in fewer iterations, and then to post-process the simulated path of the chain to reduce the variance of the resulting estimators.