Statistics and Computing

, Volume 23, Issue 5, pp 653–662 | Cite as

Zero variance Markov chain Monte Carlo for Bayesian estimators

  • Antonietta Mira
  • Reza Solgi
  • Daniele ImparatoEmail author


Interest is in evaluating, by Markov chain Monte Carlo (MCMC) simulation, the expected value of a function with respect to a, possibly unnormalized, probability distribution. A general purpose variance reduction technique for the MCMC estimator, based on the zero-variance principle introduced in the physics literature, is proposed. Conditions for asymptotic unbiasedness of the zero-variance estimator are derived. A central limit theorem is also proved under regularity conditions. The potential of the idea is illustrated with real applications to probit, logit and GARCH Bayesian models. For all these models, a central limit theorem and unbiasedness for the zero-variance estimator are proved (see the supplementary material available on-line).


Control variates GARCH models Logistic regression Metropolis-Hastings algorithm Variance reduction 



Thanks are due to D. Bressanini, for bringing to our attention the paper by Assaraf and Caffarel and helping us translate it into statistical terms; to prof. E. Regazzini and F. Nicola, for discussing the CLT conditions for the examples; P. Tenconi, F. Carone and F. Leisen for comments and contributions to a preliminary version of this research. And finally, Assaraf and Caffarel themselves have given us interesting and useful comments that have greatly improved the paper.

Supplementary material

11222_2012_9344_MOESM1_ESM.pdf (275 kb)
<Zero Variance Markov Chain Monte Carlo for Bayesian Estimators (PDF 275 kB)


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Swiss Finance InstituteUniversity of LuganoLuganoSwitzerland
  2. 2.Department of EconomicsUniversity of InsubriaVareseItaly

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