Multilevel Splitting methods, also called Sequential Monte−Carlo or Subset Simulation, are widely used methods for estimating extreme probabilities of the form where is a deterministic real-valued function and can be a random finite- or infinite-dimensional vector. Very often, is supposed to be a continuous random variable and a lot of theoretical results on the statistical behaviour of the estimator are now derived with this hypothesis. However, as soon as some threshold effect appears in and/or is discrete or mixed discrete/continuous this assumption does not hold any more and the estimator is not consistent. In this paper, we study the impact of discontinuities in the cdf of and present three unbiased corrected estimators to handle them. These estimators do not require to know in advance if is actually discontinuous or not and become all equal if is continuous. Especially, one of them has the same statistical properties in any case. Efficiency is shown on a 2-D diffusive process as well as on the Boolean SATisfiability problem (SAT).
DOI : 10.1051/ps/2015017
Mots clés : Rare event simulation, multilevel splitting, RESTART, sequential Monte Carlo, extreme event estimation, counting, Last Particle Algorithm
@article{PS_2015__19__794_0, author = {Walter, Cl\'ement}, title = {Rare event simulation and splitting for discontinuous random variables}, journal = {ESAIM: Probability and Statistics}, pages = {794--811}, publisher = {EDP-Sciences}, volume = {19}, year = {2015}, doi = {10.1051/ps/2015017}, zbl = {1416.65022}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps/2015017/} }
TY - JOUR AU - Walter, Clément TI - Rare event simulation and splitting for discontinuous random variables JO - ESAIM: Probability and Statistics PY - 2015 SP - 794 EP - 811 VL - 19 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2015017/ DO - 10.1051/ps/2015017 LA - en ID - PS_2015__19__794_0 ER -
Walter, Clément. Rare event simulation and splitting for discontinuous random variables. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 794-811. doi : 10.1051/ps/2015017. http://www.numdam.org/articles/10.1051/ps/2015017/
M. Amrein and H.R. Künsch, A variant of importance splitting for rare event estimation: Fixed number of successes. ACM Trans. Model. Comput. Simul. (TOMACS) (2011).
Estimation of small failure probabilities in high dimensions by subset simulation. Probab. Eng. Mech. 16 (2001) 263–277.
and ,A. Beskos, A. Jasra, N. Kantas and A. Thiery, On the convergence of adaptive sequential Monte Carlo methods. To appear in Ann. Appl. Probab. (2015).
Accelerating simulated annealing for the permanent and combinatorial counting problems. SIAM J. Comput. 37 (2008) 1429–1454. | Zbl
, , and ,An efficient algorithm for rare-event probability estimation, combinatorial optimization, and counting. Methodology Comput. Appl. Probab. 10 (2008) 471–505. | Zbl
and ,Efficient Monte Carlo simulation via the generalized splitting method. Stat. Comput. 22 (2012) 1–16. | Zbl
and ,Analysis of adaptive multilevel splitting algorithms in an idealized case. ESAIM: PS 19 (2015) 361–394. | Zbl
, and ,F. Cerou, P. Del Moral, A. Guyader and F. Malrieu, Fluctuation analysis of adaptive multilevel splitting. Preprint arXiv:1408.6366 (2014).
Adaptive multilevel splitting for rare event analysis. Stoch. Anal. Appl. 25 (2007) 417–443. | Zbl
and ,On the use of smoothing to improve the performance of the splitting method. Stochastic Models 27 (2011) 629–650. | Zbl
, , and ,Sequential Monte Carlo for rare event estimation. Stat. Comput. 22 (2012) 795–808. | Zbl
, , and ,B. Charles-Edouard, G. Maxime, G. Ludovic, L. Tony and R. Mathias, Unbiasedness of some generalized adaptive multilevel splitting algorithms. e-prints Preprint arXiv:1505.02674 (2015).
Sequential Monte Carlo samplers. J. Roy. Statist. Soc.: Ser. B (Statistical Methodology) 68 (2006) 411–436. | Zbl
, and ,A. Froda, Sur la Distribution des Propriétés de Voisinage des Fonctions de Variables Réelles. Ph.D. thesis, Université de Paris (1929). | JFM
M.J.J. Garvels, The splitting method in rare event simulation. Universiteit Twente (2000).
Multilevel splitting for estimating rare event probabilities. Oper. Res. 4 (1999) 585–600. | Zbl
, , and ,Simulation and estimation of extreme quantiles and extreme probabilities. Appl. Math. Optim. 64 (2011) 171–196. | Zbl
, and ,H. Hoos and T. Stiitzle, SATLlB: An online resource for research on SAT. Sat2000: highlights of satisfiability research in the year 2000 (2000) 283. | Zbl
Using TPA for Bayesian Inference. Bayesian Statistics 9 (2011) 257–282.
and ,Estimation of Particle Transmission by Random Sampling. National Bureau of Standards Applied Mathematics Series 12 (1951) 27–30.
and ,M. Mitzenmacher and E. Upfal, Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press (2005). | Zbl
R. Motwani and P. Raghavan, Randomized Algorithms. Chapman & Hall/CRC (2010). | Zbl
Monte Carlo calculation of the average extension of molecular chains. J. Chem. Phys. 23 (2004) 356–359.
and ,The Gibbs cloner for combinatorial optimization, counting and sampling. Methodol. Comput. Appl. Probab. 11 (2009) 491–549. | Zbl
,Randomized algorithms with splitting: Why the classic randomized algorithms do not work and how to make them work. Methodol. Comput. Appl. Probab. 12 (2010) 1–50. | Zbl
,The splitting method for decision making. Commun. Statistics-Simul. Comput. 41 (2012) 905–921. | Zbl
, and ,R.Y. Rubinstein and D.P. Kroese, Simulation and the Monte Carlo method. In vol. 707. John Wiley & Sons (2011). | Zbl
E. Simonnet, Combinatorial analysis of the adaptive last particle method. To appear in Stat. Comput. (2014). Doi: . | DOI
Nested sampling for general bayesian computation. Bayesian Analysis 1 (2006) 833–859. | Zbl
,Moving particles: A parallel optimal multilevel splitting method with application in quantiles estimation and meta-model based algorithms. Structural Safety 55 (2015) 10–25.
,C. Walter, Point process-based Monte Carlo estimation. To appear in Stat. Comput. (2015) Doi: . | DOI
C. Walter and G. Defaux, Rare event simulation: a point process interpretation with application in probability and quantile estimation. Proc. of the 12 International Conference on Applications of Statistics and Probability (2015).
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