Annals of Statistics

A vanilla Rao–Blackwellization of Metropolis–Hastings algorithms

Randal Douc and Christian P. Robert

Full-text: Open access


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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.

Export citation


  • Casella, G. and Robert, C. (1996). Rao-Blackwellisation of sampling schemes. Biometrika 83 81–94.
  • Casella, G. and Robert, C. (1998). Post-processing accept-reject samples: Recycling and rescaling. J. Comput. Graph. Statist. 7 139–157.
  • Delmas, J. F. and Jourdain, B. (2009). Does waste recycling really improves the multi-proposal Metropolis Hastings algorithm? An analysis based on control variates. J. Appl. Probab. 46 938–959.
  • Douc, R. and Moulines, E. (2008). Limit theorems for weighted samples with applications to sequential Monte Carlo methods. Ann. Statist. 36 2344–2376.
  • Gåsemyr, J. (2002). Markov chain Monte Carlo algorithms with independent proposal distribution and their relation to importance sampling and rejection sampling. Technical Report 2, Dept. Statistics, Univ. Oslo.
  • Latuszynski, K., Kosmidis, I., Papaspiliopoulos, O. and Roberts, G. (2010). Simulating event of unknown probabilities via reverse time martingales. Random Structures Algorithms. To appear.
  • Malefaki, S. and Iliopoulos, G. (2008). On convergence of importance sampling and other properly weighted samples to the target distribution. J. Statist. Plann. Inference 138 1210–1225.
  • Marin, J.-M. and Robert, C. (2007). Bayesian Core. Springer, New York.
  • Perron, F. (1999). Beyond accept–reject sampling. Biometrika 86 803–813.
  • Sahu, S. and Zhigljavsky, A. (1998). Adaptation for self regenerative MCMC. Technical report, Univ. Wales, Cardiff.
  • Sahu, S. and Zhigljavsky, A. (2003). Self regenerative Markov chain Monte Carlo with adaptation. Bernoulli 9 395–422.
  • Venables, W. and Ripley, B. (2002). Modern Applied Statistics with S-PLUS, 4th ed. Springer, New York.