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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
Article

Abstract

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).

Keywords

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

Notes

Acknowledgements

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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