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Zero variance Markov chain Monte Carlo for Bayesian estimators


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).

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    From a practical point of view there is no need to run two separate chains, one to get the control variates and one to get the final ZV estimator: everything can be done on a single Markov chain which is run once to estimate the optimal coefficients of the control variates and then post-processed to get the ZV estimator.


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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.

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Correspondence to Daniele Imparato.

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Mira, A., Solgi, R. & Imparato, D. Zero variance Markov chain Monte Carlo for Bayesian estimators. Stat Comput 23, 653–662 (2013). https://doi.org/10.1007/s11222-012-9344-6

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