A lognormal central limit theorem for particle approximations of normalizing constants

Jean Bérard (University of Strasbourg)
Pierre Del Moral (University of New South Wales)
Arnaud Doucet (University of Oxford)


Feynman-Kac path integration models arise in a large variety of scientic disciplines including physics, chemistry and signal processing. Their mean eld particle interpretations, termed Diusion or Quantum Monte Carlo methods in physics and Sequential Monte Carlo or Particle Filters in statistics and applied probability, have found numerous applications as they allow to sample approximately from sequences of complex probability distributions and estimate their associated normalizing constants.This article focuses on the lognormal fuctuations of these normalizing constant estimates when both the time horizon n  and the number of particles N  go to innity in such a way that n/N tends to some number between 0 and 1. To the best of our knowledge, this is the first result of this type for mean field type interacting particle systems. We also discuss special classes of models, including particle absorption models in time-homogeneous environment and hidden Markov models in ergodic random environment, for which more explicit descriptions of the limiting bias and variance can be obtained.

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Pages: 1-28

Publication Date: October 7, 2014

DOI: 10.1214/EJP.v19-3428


  • Andrieu, Christophe; Doucet, Arnaud; Holenstein, Roman. Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 (2010), no. 3, 269--342. MR2758115
  • Mathematical models and methods for ab initio quantum chemistry. Edited by M. Defranceschi and C. Le Bris. Lecture Notes in Chemistry, 74. Springer-Verlag, Berlin, 2000. xii+246 pp. ISBN: 3-540-67631-7 MR1857459
  • Cérou, F.; Del Moral, P.; Guyader, A. A nonasymptotic theorem for unnormalized Feynman-Kac particle models. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 3, 629--649. MR2841068
  • Del Moral, P.; Miclo, L. Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering. Séminaire de Probabilités, XXXIV, 1--145, Lecture Notes in Math., 1729, Springer, Berlin, 2000. MR1768060 http://archive.numdam.org/ARCHIVE/SPS/SPS_2000__34_/SPS_2000__34__1_0/SPS_2000__34__1_0.pdf http://archive.numdam.org/ARCHIVE/SPS/SPS_2000__34_/SPS_2000__34__1_0/SPS_2000__34__1_0.pdf
  • Del Moral, Pierre. Feynman-Kac formulae. Genealogical and interacting particle systems with applications. Probability and its Applications (New York). Springer-Verlag, New York, 2004. xviii+555 pp. ISBN: 0-387-20268-4 MR2044973
  • Del Moral, Pierre. Mean field simulation for Monte Carlo integration. Monographs on Statistics and Applied Probability, 126. CRC Press, Boca Raton, FL, 2013. xlvii+578 pp. ISBN: 978-1-4665-0405-9 MR3060209
  • Del Moral, Pierre; Guionnet, Alice. On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 2, 155--194. MR1819122
  • Del Moral, Pierre; Miclo, Laurent. On the stability of nonlinear Feynman-Kac semigroups. Ann. Fac. Sci. Toulouse Math. (6) 11 (2002), no. 2, 135--175. MR1988460
  • Del Moral, Pierre; Doucet, Arnaud. Particle motions in absorbing medium with hard and soft obstacles. Stochastic Anal. Appl. 22 (2004), no. 5, 1175--1207. MR2089064
  • P. Del Moral, P. Hu, and L.M. Wu. On the concentration properties of interacting particle processes. phFoundations and Trends in Machine Learning, vol. 3, no. 3--4, pp. 225--389, 2012.
  • Douc, Randal; Moulines, Eric; Olsson, Jimmy. Long-term stability of sequential Monte Carlo methods under verifiable conditions. Ann. Appl. Probab. 24 (2014), no. 5, 1767--1802. MR3226163
  • A. Doucet, M.K. Pitt, G. Deligiannidis and R. Kohn. Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator. arXiv preprint arXiv:1210.1871, 2012.
  • Sequential Monte Carlo methods in practice. Edited by Arnaud Doucet, Nando de Freitas and Neil Gordon. Statistics for Engineering and Information Science. Springer-Verlag, New York, 2001. xxviii+581 pp. ISBN: 0-387-95146-6 MR1847783
  • Durrett, Richard. Probability: theory and examples. Second edition. Duxbury Press, Belmont, CA, 1996. xiii+503 pp. ISBN: 0-534-24318-5 MR1609153
  • Grey, D. R. A note on convergence of probability measures. J. Appl. Probab. 38 (2001), no. 4, 1055--1058. MR1876558
  • L. Isserlis. On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. phBiometrika, vol. 12, pp.134-139, 1918.
  • El Makrini, Mohamed; Jourdain, Benjamin; Lelièvre, Tony. Diffusion Monte Carlo method: numerical analysis in a simple case. M2AN Math. Model. Numer. Anal. 41 (2007), no. 2, 189--213. MR2339625
  • Lelièvre, Tony; Rousset, Mathias; Stoltz, Gabriel. Free energy computations. A mathematical perspective. Imperial College Press, London, 2010. xiv+458 pp. ISBN: 978-1-84816-247-1; 1-84816-247-2 MR2681239
  • Pitt, Michael K.; Silva, Ralph dos Santos; Giordani, Paolo; Kohn, Robert. On some properties of Markov chain Monte Carlo simulation methods based on the particle filter. J. Econometrics 171 (2012), no. 2, 134--151. MR2991856
  • Rousset, Mathias. On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006), no. 3, 824--844 (electronic). MR2262944
  • C. Sherlock, A.H. Thiery, G.O. Roberts and J.S. Rosenthal. On the efficiency of pseudo-marginal random walk Metropolis algorithms. arXiv preprint arXiv:1309.7209.
  • Shiryaev, A. N. Probability. Translated from the first (1980) Russian edition by R. P. Boas. Second edition. Graduate Texts in Mathematics, 95. Springer-Verlag, New York, 1996. xvi+623 pp. ISBN: 0-387-94549-0 MR1368405
  • Whiteley, Nick. Stability properties of some particle filters. Ann. Appl. Probab. 23 (2013), no. 6, 2500--2537. MR3127943
  • Whiteley, Nick; Lee, Anthony. Twisted particle filters. Ann. Statist. 42 (2014), no. 1, 115--141. MR3178458
  • N. Whiteley and N. Kantas. A particle method for approximating principal eigen-functions and related quantities. arXiv preprint arXiv:1202.6678, 2012.
  • Wick, G. C. The evaluation of the collision matrix. Physical Rev. (2) 80, (1950). 268--272. MR0038281

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