  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. 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 idea is illustrated with real
applications to probit, logit and GARCH Bayesian models. For all these models,
a central limit theorem and unbiasedness for the zero-variance estimator are
proved (see the supplementary material available on-line).
