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Zero variance Markov chain Monte Carlo for Bayesian estimators

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).

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Notes

  1. 1.

    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.

References

  1. Adler, S.: Over-relaxation method for the Monte Carlo evaluation of the partition function for multiquadratic actions. Phys. Rev. D 23, 2901–2904 (1981)

  2. Albert, J., Chib, S.: Bayesian analysis of binary and polychotomous response data. J. Am. Stat. Assoc. 88(422), 669–679 (1993)

  3. 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)

  4. Assaraf, R., Caffarel, M.: Zero-Variance principle for Monte Carlo algorithms. Phys. Rev. Lett. 83(23), 4682–4685 (1999)

  5. 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)

  6. Barone, P., Frigessi, A.: Improving stochastic relaxation for Gaussian random fields. Probab. Eng. Inf. Sci. 4, 369–389 (1989)

  7. 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)

  8. Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econom. 31(3), 307–327 (1986)

  9. 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)

  10. Brewer, M., Aitken, C., Talbot, M.: A comparison of hybrid strategies for Gibbs sampling in mixed graphical models. Comput. Stat. 21, 343–365 (1996)

  11. Brooks, S., Gelman, A.: Some issues in monitoring convergence of iterative simulations. In: Computing Science and Statistics (1998)

  12. Craiu, R., Lemeieux, C.: Acceleration of the multiple-try Metropolis algorithm using antithetic and stratified sampling. J. Stat. Comput. 17(2), 109–120 (2007)

  13. Craiu, R., Meng, X.: Multiprocess parallel antithetic coupling for backward and forward Markov chain Monte Carlo. Ann. Stat. 33(2), 661–697 (2005)

  14. 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)

  15. Diaconis, P., Holmes, S., Neal, R.F.: Analysis of a nonreversible Markov chain sampler. Ann. Appl. Probab. 10(3), 726–752 (2000)

  16. Duane, S., Kennedy, A., Pendleton, B., Roweth, D.: Hybrid Monte Carlo Phys. Lett. B 195, 216–222 (2010)

  17. Flury, B., Riedwyl, H.: Multivariate Statistics. Chapman and Hall, London (1988)

  18. Fort, G., Moulines, E., Roberts, G., Rosenthal, S.: On the geometric ergodicity of hybrid samplers. J. Appl. Probab. 40(1), 123–146 (2003)

  19. Gelfand, A., Smith, A.: Sampling-based approaches to calculating marginal densities. J. Am. Stat. Assoc. 85, 398–409 (1990)

  20. Girolami, M., Calderhead, B.: Riemannian manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. B 73(2), 1–37 (2011)

  21. 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)

  22. Green, P.J., Mira, A.: Delayed rejection in reversible jump Metropolis-Hastings. Biometrika 88, 1035–1053 (2001)

  23. Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)

  24. Henderson, S.: Variance reduction via an approximating Markov process. Ph.D. thesis, Department of Operations Research, Stanford University, Stanford, CA (1997)

  25. Henderson, S., Glynn, P.: Approximating martingales for variance reduction in Markov process simulation. Math. Oper. Res. 27(2), 253–271 (2002)

  26. Higdon, D.: Auxiliary variable methods for Markov chain Monte Carlo with applications. J. Am. Stat. Assoc. 93, 585–595 (1998)

  27. 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)

  28. Leisen, F., Dalla Valle, L.: A new multinomial model and a zero variance estimation. Commun. Stat., Simul. Comput. 39(4), 846–859 (2010)

  29. Linnik, Y.V.: An information-theoretic proof of the central limit theorem with Lindeberg conditions. Theory Probab. Appl. 4, 288–299 (1959)

  30. Loh, W.: Methods of control variates for discrete event simulation. Ph.D. thesis, Department of Operations Research, Stanford University, Stanford, CA (1994)

  31. Marin, J.M., Robert, C.: Bayesian Core: A Practical Approach to Computational Bayesian Statistics. Springer, Berlin (2007)

  32. Mira, A., Geyer, C.J.: On reversible Markov chains. Fields Inst. Commun., Monte Carlo Methods 26, 93–108 (2000)

  33. Mira, A., Möller, J., Roberts, G.O.: Perfect slice samplers. J. R. Stat. Soc. B 63(3), 593–606 (2001)

  34. Mira, A., Tierney, L.: Efficiency and convergence properties of slice samplers. Scand. J. Stat. 29, 1–12 (2002)

  35. Neal, R.: An improved acceptance procedure for the hybrid Monte Carlo algorithm. J. Comput. Phys. 111, 194–203 (1994)

  36. Neal, R.M.: Suppressing random walks in Markov chain Monte Carlo using ordered overrelaxation. Tech. rep., Learning in Graphical Models (1995)

  37. Nelson, B.: Batch size effects on the efficiency of control variates in simulation. Eur. J. Oper. Res. 2(27), 184–196 (1989)

  38. Philippe, A., Robert, C.: Riemann sums for MCMC estimation and convergence monitoring. Stat. Comput. 11, 103–105 (2001)

  39. Ripley, B.: Stochastic Simulation. Wiley, New York (1987)

  40. 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)

  41. Swendsen, R., Wang, J.: Non universal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58, 86–88 (1987)

  42. Tierney, L.: Markov chains for exploring posterior distributions. Ann. Stat. 22, 1701–1762 (1994)

  43. Tierney, L., Mira, A.: Some adaptive Monte Carlo methods for Bayesian inference. Stat. Med. 18, 2507–2515 (1999)

  44. Van Dyk, D., Meng, X.: The art of data augmentation. J. Comput. Graph. Stat. 10, 1–50 (2001)

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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.

Author information

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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Keywords

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