A vanilla Rao--Blackwellization of Metropolis--Hastings algorithms

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.

💡 Research Summary

The paper revisits the problem of variance reduction in Metropolis–Hastings (MH) Monte Carlo estimators through Rao‑Blackwellization (RB). The classic RB approach of Casella and Robert (1996) indeed yields substantial variance reductions, but it requires storing the entire proposal trajectory and computing conditional expectations that are often costly in both time and memory. The authors therefore propose a new “vanilla” RB scheme that is universally applicable to any MH‑based estimator, incurs only a negligible computational overhead, and does not demand extra storage beyond the current state and its proposal.

The key insight is to exploit the acceptance probability α(x, y) directly. For a target function h, the ordinary MH estimator is the simple average 1/N ∑ h(Xₜ). The proposed RB estimator augments each term with a correction that accounts for the information contained in the rejected proposals: \