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).

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

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)

    Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  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)

    Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  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)

    Google Scholar 

  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)

    MATH  Google Scholar 

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

    Google Scholar 

  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)

    Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  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)

    Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

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

    MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

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

    MATH  Article  Google Scholar 

  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)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  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)

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  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)

    Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

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

    Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

Download references

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

Affiliations

Authors

Corresponding author

Correspondence to Daniele Imparato.

Electronic Supplementary Material

Below is the link to the electronic supplementary material.

<Zero Variance Markov Chain Monte Carlo for Bayesian Estimators (PDF 275 kB)

Rights and permissions

Reprints and Permissions

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

Download citation

Keywords

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