  Casella and Robert [Biometrika 83 (1996) 81--94] presented a general
Rao--Blackwellization principle for accept-reject and Metropolis--Hastings
schemes that leads to significant decreases in the variance of the resulting
estimators, but at a high cost in computation and storage. Adopting a
completely different perspective, we introduce instead a universal scheme that
guarantees variance reductions in all Metropolis--Hastings-based estimators
while keeping the computation cost under control. We establish a central limit
theorem for the improved estimators and illustrate their performances on toy
examples and on a probit model estimation.
