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An Efficient Algorithm for Rare-event Probability Estimation, Combinatorial Optimization, and Counting

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  • Zdravko I. Botev

    (University of Queensland)

  • Dirk P. Kroese

    (University of Queensland)

Abstract

Although importance sampling is an established and effective sampling and estimation technique, it becomes unstable and unreliable for high-dimensional problems. The main reason is that the likelihood ratio in the importance sampling estimator degenerates when the dimension of the problem becomes large. Various remedies to this problem have been suggested, including heuristics such as resampling. Even so, the consensus is that for large-dimensional problems, likelihood ratios (and hence importance sampling) should be avoided. In this paper we introduce a new adaptive simulation approach that does away with likelihood ratios, while retaining the multi-level approach of the cross-entropy method. Like the latter, the method can be used for rare-event probability estimation, optimization, and counting. Moreover, the method allows one to sample exactly from the target distribution rather than asymptotically as in Markov chain Monte Carlo. Numerical examples demonstrate the effectiveness of the method for a variety of applications.

Suggested Citation

  • Zdravko I. Botev & Dirk P. Kroese, 2008. "An Efficient Algorithm for Rare-event Probability Estimation, Combinatorial Optimization, and Counting," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 471-505, December.
    • Handle: RePEc:spr:metcap:v:10:y:2008:i:4:d:10.1007_s11009-008-9073-7
    DOI: 10.1007/s11009-008-9073-7
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    References listed on IDEAS

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    1. Reuven Y. Rubinstein, 2006. "How Many Needles are in a Haystack, or How to Solve #P-Complete Counting Problems Fast," Methodology and Computing in Applied Probability, Springer, vol. 8(1), pages 5-51, March.
    2. Reuven Rubinstein, 1999. "The Cross-Entropy Method for Combinatorial and Continuous Optimization," Methodology and Computing in Applied Probability, Springer, vol. 1(2), pages 127-190, September.
    3. Sigurdur Ólafsson, 2004. "Two-Stage Nested Partitions Method for Stochastic Optimization," Methodology and Computing in Applied Probability, Springer, vol. 6(1), pages 5-27, March.
    4. Dirk P. Kroese & Sergey Porotsky & Reuven Y. Rubinstein, 2006. "The Cross-Entropy Method for Continuous Multi-Extremal Optimization," Methodology and Computing in Applied Probability, Springer, vol. 8(3), pages 383-407, September.
    5. R. Y. Rubinstein, 2005. "A Stochastic Minimum Cross-Entropy Method for Combinatorial Optimization and Rare-event Estimation," Methodology and Computing in Applied Probability, Springer, vol. 7(1), pages 5-50, March.
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    Cited by:

    1. Sonal, S.D. & Ammanagi, S & Kanjilal, O & Manohar, C.S., 2018. "Experimental estimation of time variant system reliability of vibrating structures based on subset simulation with Markov chain splitting," Reliability Engineering and System Safety, Elsevier, vol. 178(C), pages 55-68.
    2. Reuven Rubinstein, 2010. "Randomized Algorithms with Splitting: Why the Classic Randomized Algorithms Do Not Work and How to Make them Work," Methodology and Computing in Applied Probability, Springer, vol. 12(1), pages 1-50, March.
    3. Paul Dupuis & Bahar Kaynar & Ad Ridder & Reuven Rubinstein & Radislav Vaisman, 2011. "Counting with Combined Splitting and Capture-Recapture Methods," Tinbergen Institute Discussion Papers 11-062/4, Tinbergen Institute.
    4. Youngjun Choe & Henry Lam & Eunshin Byon, 2018. "Uncertainty Quantification of Stochastic Simulation for Black-box Computer Experiments," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1155-1172, December.
    5. Qibin Duan & Dirk P. Kroese, 2018. "Splitting for Multi-objective Optimization," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 517-533, June.
    6. Reuven Rubinstein, 2013. "Stochastic Enumeration Method for Counting NP-Hard Problems," Methodology and Computing in Applied Probability, Springer, vol. 15(2), pages 249-291, June.
    7. Joshua Chan & Dirk Kroese, 2011. "Rare-event probability estimation with conditional Monte Carlo," Annals of Operations Research, Springer, vol. 189(1), pages 43-61, September.
    8. Hao Ma & Henk A. P. Blom, 2022. "Random Assignment Versus Fixed Assignment in Multilevel Importance Splitting for Estimating Stochastic Reach Probabilities," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2313-2338, December.
    9. Kleijnen, Jack P.C. & Ridder, A.A.N. & Rubinstein, R.Y., 2010. "Variance Reduction Techniques in Monte Carlo Methods," Other publications TiSEM 87680d1a-53c1-4107-ada4-7, Tilburg University, School of Economics and Management.
    10. Chan, Jianpeng & Papaioannou, Iason & Straub, Daniel, 2022. "An adaptive subset simulation algorithm for system reliability analysis with discontinuous limit states," Reliability Engineering and System Safety, Elsevier, vol. 225(C).
    11. Reuven Rubinstein, 2009. "The Gibbs Cloner for Combinatorial Optimization, Counting and Sampling," Methodology and Computing in Applied Probability, Springer, vol. 11(4), pages 491-549, December.
    12. Turati, Pietro & Pedroni, Nicola & Zio, Enrico, 2016. "Advanced RESTART method for the estimation of the probability of failure of highly reliable hybrid dynamic systems," Reliability Engineering and System Safety, Elsevier, vol. 154(C), pages 117-126.
    13. Ali Eshragh & Jerzy Filar & Michael Haythorpe, 2011. "A hybrid simulation-optimization algorithm for the Hamiltonian cycle problem," Annals of Operations Research, Springer, vol. 189(1), pages 103-125, September.
    14. Francesco Di Maio & Nicola Pedroni & Barnabás Tóth & Luciano Burgazzi & Enrico Zio, 2021. "Reliability Assessment of Passive Safety Systems for Nuclear Energy Applications: State-of-the-Art and Open Issues," Energies, MDPI, vol. 14(15), pages 1-17, August.
    15. Radislav Vaisman & Dirk P. Kroese, 2017. "Stochastic Enumeration Method for Counting Trees," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 31-73, March.
    16. M. Garvels, 2011. "A combined splitting—cross entropy method for rare-event probability estimation of queueing networks," Annals of Operations Research, Springer, vol. 189(1), pages 167-185, September.

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