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).
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
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.
Adler, S.: Over-relaxation method for the Monte Carlo evaluation of the partition function for multiquadratic actions. Phys. Rev. D 23, 2901–2904 (1981)
Albert, J., Chib, S.: Bayesian analysis of binary and polychotomous response data. J. Am. Stat. Assoc. 88(422), 669–679 (1993)
Ardia, D.: Financial Risk Management with Bayesian Estimation of GARCH Models: Theory and Applications. Lecture Notes in Economics and Mathematical Systems, vol. 612. Springer, Berlin (2008)
Assaraf, R., Caffarel, M.: Zero-Variance principle for Monte Carlo algorithms. Phys. Rev. Lett. 83(23), 4682–4685 (1999)
Assaraf, R., Caffarel, M.: Zero-variance zero-bias principle for observables in quantum Monte Carlo: application to forces. J. Chem. Phys. 119(20), 10,536–10,552 (2003)
Barone, P., Frigessi, A.: Improving stochastic relaxation for Gaussian random fields. Probab. Eng. Inf. Sci. 4, 369–389 (1989)
Barone, P., Sebastiani, G., Stander, J.: General over-relaxation Markov chain Monte Carlo algorithms for Gaussian densities. Stat. Probab. Lett. 52(2), 115–124 (2001)
Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econom. 31(3), 307–327 (1986)
Bollerslev, T.: Glossary to ARCH (GARCH). In: Bollerslev, T., Russell, J., Watson, M. (eds.) Volatility and Time Series Econometrics, Essays in Honor of Robert Engle. Oxford University Press, Oxford (2010)
Brewer, M., Aitken, C., Talbot, M.: A comparison of hybrid strategies for Gibbs sampling in mixed graphical models. Comput. Stat. 21, 343–365 (1996)
Brooks, S., Gelman, A.: Some issues in monitoring convergence of iterative simulations. In: Computing Science and Statistics (1998)
Craiu, R., Lemeieux, C.: Acceleration of the multiple-try Metropolis algorithm using antithetic and stratified sampling. J. Stat. Comput. 17(2), 109–120 (2007)
Craiu, R., Meng, X.: Multiprocess parallel antithetic coupling for backward and forward Markov chain Monte Carlo. Ann. Stat. 33(2), 661–697 (2005)
Dellaportas, P., Kontoyiannis, I.: Control variates for estimation based on reversible Markov chain Monte Carlo samplers. J. R. Stat. Soc. B 74(1), 133–161 (2012)
Diaconis, P., Holmes, S., Neal, R.F.: Analysis of a nonreversible Markov chain sampler. Ann. Appl. Probab. 10(3), 726–752 (2000)
Duane, S., Kennedy, A., Pendleton, B., Roweth, D.: Hybrid Monte Carlo Phys. Lett. B 195, 216–222 (2010)
Flury, B., Riedwyl, H.: Multivariate Statistics. Chapman and Hall, London (1988)
Fort, G., Moulines, E., Roberts, G., Rosenthal, S.: On the geometric ergodicity of hybrid samplers. J. Appl. Probab. 40(1), 123–146 (2003)
Gelfand, A., Smith, A.: Sampling-based approaches to calculating marginal densities. J. Am. Stat. Assoc. 85, 398–409 (1990)
Girolami, M., Calderhead, B.: Riemannian manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. B 73(2), 1–37 (2011)
Green, P., Han, X.: Metropolis methods, Gaussian proposals, and antithetic variables. In: Barone, P., Frigessi, A., Piccioni, M. (eds.) Lecture Notes in Statistics, Stochastic Methods and Algorithms in Image Analysis, vol. 74, pp. 142–164. Springer, Berlin (1992)
Green, P.J., Mira, A.: Delayed rejection in reversible jump Metropolis-Hastings. Biometrika 88, 1035–1053 (2001)
Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)
Henderson, S.: Variance reduction via an approximating Markov process. Ph.D. thesis, Department of Operations Research, Stanford University, Stanford, CA (1997)
Henderson, S., Glynn, P.: Approximating martingales for variance reduction in Markov process simulation. Math. Oper. Res. 27(2), 253–271 (2002)
Higdon, D.: Auxiliary variable methods for Markov chain Monte Carlo with applications. J. Am. Stat. Assoc. 93, 585–595 (1998)
Ishwaran, H.: Applications of hybrid Monte Carlo to Bayesian generalized linear models: quasicomplete separation and neural networks. J. Comput. Graph. Stat. 8, 779–799 (1999)
Leisen, F., Dalla Valle, L.: A new multinomial model and a zero variance estimation. Commun. Stat., Simul. Comput. 39(4), 846–859 (2010)
Linnik, Y.V.: An information-theoretic proof of the central limit theorem with Lindeberg conditions. Theory Probab. Appl. 4, 288–299 (1959)
Loh, W.: Methods of control variates for discrete event simulation. Ph.D. thesis, Department of Operations Research, Stanford University, Stanford, CA (1994)
Marin, J.M., Robert, C.: Bayesian Core: A Practical Approach to Computational Bayesian Statistics. Springer, Berlin (2007)
Mira, A., Geyer, C.J.: On reversible Markov chains. Fields Inst. Commun., Monte Carlo Methods 26, 93–108 (2000)
Mira, A., Möller, J., Roberts, G.O.: Perfect slice samplers. J. R. Stat. Soc. B 63(3), 593–606 (2001)
Mira, A., Tierney, L.: Efficiency and convergence properties of slice samplers. Scand. J. Stat. 29, 1–12 (2002)
Neal, R.: An improved acceptance procedure for the hybrid Monte Carlo algorithm. J. Comput. Phys. 111, 194–203 (1994)
Neal, R.M.: Suppressing random walks in Markov chain Monte Carlo using ordered overrelaxation. Tech. rep., Learning in Graphical Models (1995)
Nelson, B.: Batch size effects on the efficiency of control variates in simulation. Eur. J. Oper. Res. 2(27), 184–196 (1989)
Philippe, A., Robert, C.: Riemann sums for MCMC estimation and convergence monitoring. Stat. Comput. 11, 103–105 (2001)
Ripley, B.: Stochastic Simulation. Wiley, New York (1987)
So, M.K.P.: Bayesian analysis of nonlinear and non-Gaussian state space models via multiple-try sampling methods. Stat. Comput. 16, 125–141 (2006)
Swendsen, R., Wang, J.: Non universal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58, 86–88 (1987)
Tierney, L.: Markov chains for exploring posterior distributions. Ann. Stat. 22, 1701–1762 (1994)
Tierney, L., Mira, A.: Some adaptive Monte Carlo methods for Bayesian inference. Stat. Med. 18, 2507–2515 (1999)
Van Dyk, D., Meng, X.: The art of data augmentation. J. Comput. Graph. Stat. 10, 1–50 (2001)
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.
Electronic Supplementary Material
Below is the link to the electronic supplementary material.
About this article
Cite this article
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
- Control variates
- GARCH models
- Logistic regression
- Metropolis-Hastings algorithm
- Variance reduction