Antithetic methods for Gibbs samplers

In this article we propose a modification to the output fromMetropolis-within-Gibbs samplers that can lead to substantial reductions in the variance over standard estimates. The idea is simple: at each time step of the algorithm, introduce an extra sample into the estimate that is negatively correlated with the current sample, the rationale being that this provides a two-sample numerical approximation to a Rao-Blackwellized estimate. As the conditional sampling distribution at each step has already been constructed, the generation of the antithetic sample often requires negligible computational effort. Our method is implementable whenever one subvector of the state can be sampled from its full conditional and the corresponding distribution function may be inverted, or the full conditional has a symmetric density. We demonstrate our approach in the context of logistic regression and hierarchical Poisson models. The data and computer code used in this article are available online. © 2009 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America.