Abstract
We study approximations of evolving probability measures by an interacting particle system. The particle system dynamics is a combination of independent Markov chain moves and importance sampling/resampling steps. Under global regularity conditions, we derive non-asymptotic error bounds for the particle system approximation. In a few simple examples, including high dimensional product measures, bounds with explicit constants of feasible size are obtained. Our main motivation are applications to sequential MCMC methods for Monte Carlo integral estimation.
References
Bengtsson, Th., Bickel, P., Li, B.: Curse-of-dimensionality revisited: collapse of the particle filter in very large scale systems. In: Probability and statistics: essays in honor of David A. Freedman, Inst. Math. Stat. Collect., vol. 2, pp. 316–334. Inst. Math. Statist., Beachwood, OH (2008)
Beskos A., Crisan D., Jasra A.: On the stability of a class of sequential Monte Carlo methods in high dimensions. Technical report, Imperial College, London (2011)
Bickel, P., Li, Bo., Bengtsson, Th.: Sharp failure rates for the bootstrap particle filter in high dimensions. In: Pushing the limits of contemporary statistics: contributions in honor of Jayanta K. Ghosh, Inst. Math. Stat. Collect., vol. 3, pp. 318–329. Inst. Math. Statist., Beachwood, OH (2008)
Cappé O., Guillin A., Marin J.M., Robert C.P.: Population Monte Carlo. J. Comput. Graph. Stat. 13(4), 907–929 (2004)
Cappé O., Moulines E., Rydén T.: Inference in hidden Markov models. Springer Series in Statistics, Springer, New York (2005)
Cérou F., Del Moral P., Guyader A.: A nonasymptotic theorem for unnormalized Feynman–Kac particle models. Ann. Inst. H. Poincaré Probab. Stat. 47(3), 629–649 (2011)
Chopin N.: A sequential particle filter method for static models. Biometrika 89(3), 539–551 (2002)
Chopin N.: Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Stat. 32(6), 2385–2411 (2004)
Del Moral P.: Feynman–Kac formulae. Springer-Verlag, New York (2004)
Del Moral P., Doucet A., Jasra A.: Sequential Monte Carlo samplers. J. R. Stat. Soc. B 68(3), 411–436 (2006)
Del Moral, P., Doucet, A., Jasra, A.: On adaptive resampling procedures for sequential Monte Carlo methods. Bernoulli (in press)
Del Moral P., Guionnet A.: On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré Probab. Stat. 37(2), 155–194 (2001)
Del Moral P., Miclo L.: On the convergence and applications of generalized simulated annealing. SIAM J. Control Optim. 37(4), 1222–1250 (1999) (electronic)
Del Moral, P., Miclo, L.: Branching and interacting particle systems approximations of Feynman–Kac formulae with applications to non-linear filtering. In: Séminaire de Probabilités, XXXIV, Lecture Notes in Math., vol. 1729, pp. 1–145. Springer, Berlin (2000)
Diaconis P.: The cutoff phenomenon in finite Markov chains. Proc. Nat. Acad. Sci. USA 93(4), 1659–1664 (1996)
Ding J., Lubetzky E., Peres Y.: Total variation cutoff in birth-and-death chains. Probab. Theory Relat. Fields 146(1–2), 61–85 (2010)
Douc R., Guillin R., Marin J.-M., Robert C.P.: Minimum variance importance sampling via population Monte Carlo. ESAIM Probab. Stat. 11, 427–447 (2007) (electronic)
Douc R., Moulines E.: Limit theorems for weighted samples with applications to sequential Monte Carlo methods. Ann. Stat. 36(5), 2344–2376 (2008)
Doucet, A., Freitas, N., Gordon, N. (eds): Sequential Monte Carlo methods in practice. Springer-Verlag, New York (2001)
Eberle, A., Marinelli, C.: Stability of sequential Markov chain Monte Carlo methods. In: Conference Oxford sur les méthodes de Monte Carlo séquentielles, ESAIM Proc., vol. 19, pp. 22–31. EDP Sci., Les Ulis (2007)
Eberle A., Marinelli C.: L p estimates for Feynman–Kac propagators with time-dependent reference measures. J. Math. Anal. Appl. 365(1), 120–134 (2010)
Geyer, C.J.: Markov chain Monte Carlo maximum likelihood. In: Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, pp. 156–163 (1991)
Gīhman Ĭ.Ī., Skorohod A.V.: The theory of stochastic processes. II. Springer-Verlag, New York (1975)
Gulisashvili A., van Casteren J.A.: Non-autonomous Kato classes and Feynman–Kac propagators. World Scientific Publishing, Hackensack NJ (2006)
Hukushima K., Nemoto K.: Exchange Monte Carlo method and application to spin glass simulations. J. Phys. Soc. Jpn. 65(6), 1604–1608 (1996)
Jarzynski C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78(14), 2690–2693 (1997)
Jasra A., Doucet A.: Stability of sequential Monte Carlo samplers via the Foster–Lyapunov condition. Statist. Probab. Lett. 78(17), 3062–3069 (2008)
Kipnis C., Landim C.: Scaling limits of interacting particle systems. Springer-Verlag, Berlin (1999)
Kou S.C., Zhou Q., Wong W.H.: Equi-energy sampler with applications in statistical inference and statistical mechanics. Ann. Stat. 34(4), 1581–1652 (2006)
Liu J.S.: Monte Carlo strategies in scientific computing. Springer-Verlag, New York (2001)
Madras N., Zheng Z.: On the swapping algorithm. Random Struct. Algorithms 22(1), 66–97 (2003)
Miclo L.: An example of application of discrete Hardy’s inequalities. Markov Process. Relat. Fields 5(3), 319–330 (1999)
Neal R.M.: Annealed importance sampling. Stat. Comput. 11(2), 125–139 (2001)
Robert C.P., Casella G.: Monte Carlo statistical methods, 2nd edn. Springer-Verlag, New York (2004)
Rousset, M.: Continuous time population Monte Carlo and computational physics, Ph.D. thesis, Université Paul Sabatier, Toulouse (2006)
Rousset M.: On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38(3), 824–844 (2006) (electronic)
Saloff-Coste, L.: Lectures on finite Markov chains, Lectures on probability theory and statistics (Saint-Flour, 1996). In: Lecture Notes in Math., vol. 1665, pp. 301–413. Springer, Berlin (1997)
Schweizer, N.: Non-asymptotic error bounds for Sequential MCMC methods. Ph.D. thesis, Universität Bonn (2011)
Stannat W.: On the convergence of genetic algorithms—a variational approach. Probab. Theory Relat. Fields 129(1), 113–132 (2004)
Whiteley, N.: Sequential Monte Carlo samplers: error bounds and insensitivity to initial conditions. arXiv:1103.3970v1
Author information
Authors and Affiliations
Corresponding author
Additional information
We would like to thank the anonymous referees for detailed reports and very helpful comments on the first version of this article. This work was partially supported by the Sonderforschungsbereich 611, Bonn. The second-named author also gratefully acknowledges the support of the DAAD.
Rights and permissions
About this article
Cite this article
Eberle, A., Marinelli, C. Quantitative approximations of evolving probability measures and sequential Markov chain Monte Carlo methods. Probab. Theory Relat. Fields 155, 665–701 (2013). https://doi.org/10.1007/s00440-012-0410-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-012-0410-y
Keywords
- Markov chain Monte Carlo
- Sequential Monte Carlo
- Importance sampling
- Spectral gap
- Dirichlet forms
- Functional inequalities
- Feynman–Kac formula
Mathematics Subject Classification (2000)
- 65C05
- 60J25
- 60B10
- 47H20
- 47D08