A nonasymptotic theorem for unnormalized Feynman-Kac particle models
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 3, p. 629-649

We present a nonasymptotic theorem for interacting particle approximations of unnormalized Feynman-Kac models. We provide an original stochastic analysis-based on Feynman-Kac semigroup techniques combined with recently developed coalescent tree-based functional representations of particle block distributions. We present some regularity conditions under which the -relative error of these weighted particle measures grows linearly with respect to the time horizon yielding what seems to be the first results of this type for this class of unnormalized models. We also illustrate these results in the context of particle absorption models, with a special interest in rare event analysis.

Nous présentons un théorème non asymptotique pour les approximation par systèmes de particules en interaction des modèles de Feynman-Kac non normalisés. Nous introduisons une analyse stochastique originale basée sur des techniques de semigroupes de Feynman-Kac, associées avec les représentation, récemment proposées, des distributions de blocks de particules, en terme de développement en arbre de coalescence. Nous présentons des conditions de régularité sous lesquelles l'erreur relative de ces mesures particulaires pondérées croît linéairement par rapport ‘a l'horizon temporel, conduisant ‘a ce qui semble être le premier résultat de ce type pour cette classe de modèles non normalisés. Nous illustrons ces résultats dans le contexte des mesures statiques de Boltzmann-Gibbs et des distributions restreintes, avec un intérêt particulier pour les événements rares.

DOI : https://doi.org/10.1214/10-AIHP358
Classification:  47D08,  60C05,  60K35,  65C35,  31B10,  60J80,  65C05,  92D25
Keywords: interacting particle systems, Feynman-Kac semigroups, nonasymptotic estimates, genetic algorithms, Boltzmann-Gibbs measures, Monte Carlo models, rare events
@article{AIHPB_2011__47_3_629_0,
     author = {C\'erou, F. and Del Moral, P. and Guyader, A.},
     title = {A nonasymptotic theorem for unnormalized Feynman-Kac particle models},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {3},
     year = {2011},
     pages = {629-649},
     doi = {10.1214/10-AIHP358},
     zbl = {1233.60047},
     mrnumber = {2841068},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_3_629_0}
}
Cérou, F.; Del Moral, P.; Guyader, A. A nonasymptotic theorem for unnormalized Feynman-Kac particle models. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 3, pp. 629-649. doi : 10.1214/10-AIHP358. http://www.numdam.org/item/AIHPB_2011__47_3_629_0/

[1] J. H. Blanchet, P. W. Glynn, P. L'Ecuyer, W. Sandmann and B. Tuffin. Asymptotic robustness of estimators in rare-event simulation. In Proc. of the 2007 INFORMS Workshop. Fontainebleau, France, 2007. Available at http://www.irisa.fr/dionysos/pages_perso/tuffin/Publis/robust-informs06.pdf.

[2] F. Cerou, P. Del Moral, F. Le Gland and P. Lezaud. Genealogical models in entrance times rare event analysis. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 181-203. | MR 2249654 | Zbl 1104.60044

[3] F. Cerou and A. Guyader. Adaptive multilevel splitting for rare event analysis. Stoch. Anal. Appl. 25 (2007) 417-433. | MR 2303095 | Zbl 1220.65009

[4] T. Dean and P. Dupuis. Splitting for rare event simulation: A large deviations approach to design and analysis. Stochastic Process. Appl. 119 (2009) 562-587. | MR 2494004 | Zbl 1157.60019

[5] P. Del Moral. Feynman-Kac Formulae. Genealogical and Interacting Particle Systems. Springer, New York, 2004. | MR 2044973 | Zbl 1130.60003

[6] P. Del Moral, A. Doucet and A. Jasra. Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 (2006) 411-436. | MR 2278333 | Zbl 1105.62034

[7] P. Del Moral, A. Doucet and G. W. Peters. Sharp propagations of chaos estimates for Feynman-Kac particle models. Theory Probab. Appl. 51 (2007) 459-485. | MR 2325545 | Zbl 1156.60072

[8] P. Del Moral, F. Patras and S. Rubenthaler. Coalescent tree based functional representations for some Feynman-Kac particle models. Ann. Appl. Probab. 19 (2009) 778-825. | MR 2521888 | Zbl 1189.60171

[9] A. Doucet, N. De Freitas and N. Gordon, eds. Sequential Monte Carlo Methods in Practice. Springer, New York, 2001. | MR 1847783 | Zbl 1056.93576

[10] A. M. Johansen, P. Del Moral and A. Doucet. Sequential Monte Carlo samplers for rare events. In Proceedings of 6th International Workshop on Rare Event Simulation. Bamberg, Germany, 2006.

[11] A. Lagnoux. Rare event simulation. Probab. Engrg. Inform. Sci. 20 (2006) 45-66. | MR 2187629 | Zbl 1101.65005