The Annals of Applied Probability

Fluctuation analysis of adaptive multilevel splitting

Frédéric Cérou and Arnaud Guyader

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Multilevel Splitting, also called Subset Simulation, is a Sequential Monte Carlo method to simulate realisations of a rare event as well as to estimate its probability. This article is concerned with the convergence and the fluctuation analysis of Adaptive Multilevel Splitting techniques. In contrast to their fixed level version, adaptive techniques estimate the sequence of levels on the fly and in an optimal way, with only a low additional computational cost. However, very few convergence results are available for this class of adaptive branching models, mainly because the sequence of levels depends on the occupation measures of the particle systems. This article proves the consistency of these methods as well as a central limit theorem. In particular, we show that the precision of the adaptive version is the same as the one of the fixed-levels version where the levels would have been placed in an optimal manner.

Article information

Ann. Appl. Probab., Volume 26, Number 6 (2016), 3319-3380.

Received: December 2014
Revised: September 2015
First available in Project Euclid: 15 December 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C35: Stochastic particle methods [See also 82C80] 65C05: Monte Carlo methods
Secondary: 60J85: Applications of branching processes [See also 92Dxx] 47D08: Schrödinger and Feynman-Kac semigroups

Sequential Monte Carlo rare events interacting particle systems Feynman–Kac semigroups


Cérou, Frédéric; Guyader, Arnaud. Fluctuation analysis of adaptive multilevel splitting. Ann. Appl. Probab. 26 (2016), no. 6, 3319--3380. doi:10.1214/16-AAP1177.

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  • [1] Au, S. K. and Beck, J. L. (2001). Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics 16 263–277.
  • [2] Au, S. K. and Beck, J. L. (2003). Subset simulation and its application to seismic risk based on dynamic analysis. Journal of Engineering Mechanics 129 901–917.
  • [3] Beskos, A., Jasra, A., Kantas, N. and Thiery, A. (2016). On the convergence of adaptive sequential Monte Carlo methods. Ann. Appl. Probab. 26 1111–1146.
  • [4] Biau, G., Cérou, F. and Guyader, A. (2015). New insights into approximate Bayesian computation. Ann. Inst. Henri Poincaré Probab. Stat. 51 376–403.
  • [5] Botev, Z. I. and Kroese, D. P. (2008). An efficient algorithm for rare-event probability estimation, combinatorial optimization, and counting. Methodol. Comput. Appl. Probab. 10 471–505.
  • [6] Bréhier, C.-E., Lelièvre, T. and Rousset, M. (2015). Analysis of adaptive multilevel splitting algorithms in an idealized case. ESAIM Probab. Stat. 19 361–394.
  • [7] Bucklew, J. A. (2004). Introduction to Rare Event Simulation. Springer, New York.
  • [8] Cérou, F., Del Moral, P., Furon, T. and Guyader, A. (2012). Sequential Monte Carlo for rare event estimation. Stat. Comput. 22 795–808.
  • [9] Cérou, F., Del Moral, P. and Guyader, A. (2011). A nonasymptotic theorem for unnormalized Feynman–Kac particle models. Ann. Inst. Henri Poincaré Probab. Stat. 47 629–649.
  • [10] Cérou, F., Del Moral, P., Le Gland, F. and Lezaud, P. (2006). Genetic genealogical models in rare event analysis. ALEA Lat. Am. J. Probab. Math. Stat. 1 181–203.
  • [11] Cérou, F. and Guyader, A. (2007). Adaptive multilevel splitting for rare event analysis. Stoch. Anal. Appl. 25 417–443.
  • [12] Chopin, N. (2004). Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 2385–2411.
  • [13] Del Moral, P. (2004). Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, New York.
  • [14] Del Moral, P. (2013). Mean Field Simulation for Monte Carlo Integration. Monographs on Statistics and Applied Probability 126. CRC Press, Boca Raton, FL.
  • [15] Del Moral, P., Doucet, A. and Jasra, A. (2006). Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 411–436.
  • [16] Del Moral, P., Doucet, A. and Jasra, A. (2012). On adaptive resampling strategies for sequential Monte Carlo methods. Bernoulli 18 252–278.
  • [17] Del Moral, P. and Miclo, L. (2000). 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. 1729 1–145. Springer, Berlin.
  • [18] Devroye, L. (1986). Lecture Notes on Bucket Algorithms. Birkhäuser, Boston, MA.
  • [19] Devroye, L., Györfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition. Springer, New York.
  • [20] Douc, R. and Moulines, E. (2008). Limit theorems for weighted samples with applications to sequential Monte Carlo methods. Ann. Statist. 36 2344–2376.
  • [21] Evans, L. C. and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL.
  • [22] Giraud, F. and Del Moral, P. (2016). Non-asymptotic analysis of adaptive and annealed Feynman–Kac particle models. Bernoulli. To appear.
  • [23] Glasserman, P. and Wang, Y. (1997). Counterexamples in importance sampling for large deviations probabilities. Ann. Appl. Probab. 7 731–746.
  • [24] Gobet, E. and Liu, G. (2015). Rare event simulation using reversible shaking transformations. SIAM J. Sci. Comput. 37 A2295–A2316.
  • [25] Guyader, A., Hengartner, N. and Matzner-Løber, E. (2011). Simulation and estimation of extreme quantiles and extreme probabilities. Appl. Math. Optim. 64 171–196.
  • [26] Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97–109.
  • [27] Kahn, H. and Harris, T. E. (1951). Estimation of particle transmission by random sampling. National Bureau of Standards Appl. Math. Series 12 27–30.
  • [28] L’Ecuyer, P., Le Gland, F., Lezaud, P. and Tuffin, B. (2009). Splitting techiques. In Rare event simulation using Monte Carlo methods (G. Rubino and B. Tuffin, eds.) 39–61. Wiley, Chichester.
  • [29] L’Ecuyer, P., Mandjes, M. and Tuffin, B. (2009). Importance sampling in rare event simulation. In Rare event simulation using Monte Carlo methods (G. Rubino and B. Tuffin, eds.) 17–38. Wiley, Chichester.
  • [30] Legoll, F. and Lelièvre, T. (2010). Effective dynamics using conditional expectations. Nonlinearity 23 2131–2163.
  • [31] Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equation of state calculations by fast computing machines. Journal of Chemical Physics 21 1087–1092.
  • [32] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • [33] Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York.
  • [34] Rosenbluth, M. N. and Rosenbluth, A. W. (1955). Monte Carlo calculation of the average extension of molecular chains. Journal of Chemical Physics 23 356–359.
  • [35] Rubinstein, R. (2009). The Gibbs cloner for combinatorial optimization, counting and sampling. Methodol. Comput. Appl. Probab. 11 491–549.
  • [36] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • [37] Simonnet, E. (2016). Combinatorial analysis of the adaptive last particle method. Statistics and Computing 26 211–230.
  • [38] Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22 1701–1762.