Antithetic Coupling of Two Gibbs Sampler Chains

Two coupled Gibbs sampler chains, both with invariant probability density , are run in parallel in such a way that the chains are negatively correlated. This allows us to define an estimator of the expectation E with respect to which achieves significant variance reduction with respect to the usual Gibbs sampler at comparable computational costs. We show that the asymptotic variance of the estimator based on the new algorithm is always smaller than the variance of a single Gibbs sampler chain, if is either attractive or repulsive and is componentwise monotone. The new antithetic algorithm is shown to outperform the standard Gibbs sampler by one order of magnitude when is a multivariate normal density or the Ising model. Numerical experiments show that the antithetically coupled Gibbs samplers reduce the finite sample variance in several other models to less than one third, often one fifth, when run for the same time.