Zero variance Markov chain Monte Carlo for Bayesian estimators
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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).
KeywordsControl 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.
- Brooks, S., Gelman, A.: Some issues in monitoring convergence of iterative simulations. In: Computing Science and Statistics (1998) Google Scholar
- Duane, S., Kennedy, A., Pendleton, B., Roweth, D.: Hybrid Monte Carlo Phys. Lett. B 195, 216–222 (2010) Google Scholar
- Henderson, S.: Variance reduction via an approximating Markov process. Ph.D. thesis, Department of Operations Research, Stanford University, Stanford, CA (1997) Google Scholar
- Loh, W.: Methods of control variates for discrete event simulation. Ph.D. thesis, Department of Operations Research, Stanford University, Stanford, CA (1994) Google Scholar
- Marin, J.M., Robert, C.: Bayesian Core: A Practical Approach to Computational Bayesian Statistics. Springer, Berlin (2007) Google Scholar
- Neal, R.M.: Suppressing random walks in Markov chain Monte Carlo using ordered overrelaxation. Tech. rep., Learning in Graphical Models (1995) Google Scholar