Zero variance Markov chain Monte Carlo for Bayesian estimators

A general purpose variance reduction technique for Markov chain Monte Carlo (MCMC) estimators, based on the zero-variance principle introduced in the physics literature, is proposed to evaluate the expected value, μf , of a function f with respect to a, possibly unnormalized, probability distribution π. In this context, a control variate approach, generally used for Monte Carlo simulation, is exploited by replacing f with a different function, f̃ . The function f̃ is constructed so that its expectation, under π, equals μf , but its variance with respect to π is much smaller. Theoretically, an optimal re-normalization μf exists which may lead to zero variance; in practice, a suitable approximation for it must be investigated. In this paper, an efficient class of re-normalized f̃ is investigated, based on a polynomial parametrization. We find that a low-degree polynomial (1st, 2nd or 3rd degree) can lead to dramatically huge variance reduction of the resulting zero-variance MCMC estimator. General formulas for the construction of the control variates in this context are given. These allow for an easy implementation of the method in very general settings regardless of the form of the target/posterior distribution (only differentiability is required) and of the MCMC algorithm implemented (in particular, no reversibility is needed). ∗Università dell’Insubria, Varese. E-mail: [email protected] †Istituto di Finanza, Università di Lugano. E-mail: [email protected] ‡Università dell’Insubria, Varese. E-mail: [email protected]