The Annals of Statistics

A vanilla Rao–Blackwellization of Metropolis–Hastings algorithms

Randal Douc and Christian P. Robert

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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.

Article information

Ann. Statist., Volume 39, Number 1 (2011), 261-277.

First available in Project Euclid: 3 December 2010

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Primary: 62-04: Explicit machine computation and programs (not the theory of computation or programming) 60F05: Central limit and other weak theorems 60J22: Computational methods in Markov chains [See also 65C40] 60J05: Discrete-time Markov processes on general state spaces 62B10: Information-theoretic topics [See also 94A17]

Metropolis–Hastings algorithm Markov chain Monte Carlo (MCMC) probit model central limit theorem variance reduction conditioning


Douc, Randal; Robert, Christian P. A vanilla Rao–Blackwellization of Metropolis–Hastings algorithms. Ann. Statist. 39 (2011), no. 1, 261--277. doi:10.1214/10-AOS838.

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