Skip to main content
Log in

Markov chain importance sampling with applications to rare event probability estimation

  • Published:
Statistics and Computing Aims and scope Submit manuscript

Abstract

We present a versatile Monte Carlo method for estimating multidimensional integrals, with applications to rare-event probability estimation. The method fuses two distinct and popular Monte Carlo simulation methods—Markov chain Monte Carlo and importance sampling—into a single algorithm. We show that for some applied numerical examples the proposed Markov Chain importance sampling algorithm performs better than methods based solely on importance sampling or MCMC.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algorithms and Analysis. Springer, New York (2007)

    MATH  Google Scholar 

  • Asmussen, S., Blanchet, J., Juneja, S., Rojas-Nandayapa, L.: Efficient simulation of tail probabilities of sums of correlated lognormals. Ann. Oper. Res. (2009). doi:10.1007/s10479-009-0658-5

    MATH  Google Scholar 

  • Bassamboo, A., Juneja, S., Zeevi, A.: Portfolio credit risk with extremal dependence: Asymptotic analysis and efficient simulation. Oper. Res. 56(3), 593–606 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Besag, J.: Statistical analysis of non-lattice data. Statistician 24(3), 179–195 (1975)

    Article  MathSciNet  Google Scholar 

  • Blanchet, J.H., Glynn, P.W.: Efficient rare-event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Probab. 18, 1351–1378 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Botev, Z.I., Kroese, D.P.: Efficient Monte Carlo simulation via the generalized splitting method. Stat. Comput. (2010). doi:10.1007/s11222-010-9201-4

    Google Scholar 

  • Botev, Z.I., L’Ecuyer, P., Tuffin, B.: Importance sampling method based on a one-step look-ahead density from a Markov chain. In: Proceedings of the 2011 Winter Simulation Conference, Phoenix, AZ (2011)

    Google Scholar 

  • Botev, Z.I., L’Ecuyer, P., Rubino, G., Simard, R., Tuffin, B.: Static network reliability estimation via generalized splitting. INFORMS Journal on Computing (2012, to appear)

  • Brereton, T.J., Chan, J.C.C., Kroese, D.P.: Fitting mixture importance sampling distributions via improved cross-entropy. In: Proceedings of the 2011 Winter Simulation Conference, Phoenix, AZ (2011)

    Google Scholar 

  • Bucklew, J.: Introduction to Rare-Event Simulation. Springer, New York (2004)

    MATH  Google Scholar 

  • Cancela, H., El Khadiri, M., Rubino, G.: Rare event analysis by Monte Carlo techniques in static models. In: Rubino, G., Tuffin, B. (eds.) Rare Event Simulation Using Monte Carlo Methods, pp. 145–170. Wiley, New York (2009a). Chap. 7

    Chapter  Google Scholar 

  • Cancela, H., L’Ecuyer, P., Lee, M., Rubino, G., Tuffin, B.: Analysis and improvements of path-based methods for Monte Carlo reliability evaluation of static models. In: Faulin, J., Juan, A.A., Martorell, S., Ramirez-Marquez, E. (eds.) Simulation Methods for Reliability and Availability of Complex Systems, pp. 65–84. Springer, Berlin (2009b)

    Google Scholar 

  • Chan, J.C.C., Kroese, D.P.: Improved cross-entropy method for estimation. Stat. Comput. (2011). doi:10.1007/s11222-011-9275-7

    Google Scholar 

  • Chan, J.C.C., Glynn, P.W., Kroese, D.P.: A comparison of cross-entropy and variance minimization strategies. J. Appl. Probab. 48A, 183–194 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Chib, S.: Marginal likelihood from the Gibbs output. J. Am. Stat. Assoc. 90(432), 1313–1321 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Dupuis, P., Leder, K., Wang, H.: Importance sampling for sums of random variables with regularly varying tails. ACM Trans. Model. Comput. Simul. 17(3), 14 (2006)

    Article  Google Scholar 

  • Gertsbakh, I.B., Shpungin, Y.: Models of Network Reliability. CRC Press, Boca Raton (2010)

    MATH  Google Scholar 

  • Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, New York (2004)

    MATH  Google Scholar 

  • Glasserman, P., Wang, Y.: Counterexamples in importance sampling for large deviations probabilities. Ann. Appl. Probab. 7(3), 731–746 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Kroese, D.P., Taimre, T., Botev, Z.I.: Handbook of Monte Carlo Methods. Wiley, New York (2011)

    Book  MATH  Google Scholar 

  • L’Ecuyer, P., Tuffin, B.: Approximate zero-variance simulation. In: Proceedings of the 2008 Winter Simulation Conference, pp. 170–181. IEEE Press, New York (2008)

    Chapter  Google Scholar 

  • L’Ecuyer, P., Blanchet, J.H., Tuffin, B., Glynn, P.W.: Asymptotic robustness of estimators in rare-event simulation. ACM Trans. Model. Comput. Simul. 20(1), 6 (2010)

    Google Scholar 

  • Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer, New York (2001)

    MATH  Google Scholar 

  • Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953)

    Article  Google Scholar 

  • Philippe, A.: Simulation of right and left truncated gamma distribution by mixtures. Stat. Comput. 7, 173–181 (1997)

    Article  Google Scholar 

  • Robert, C.P., Casella, G.: Monte Carlo Statistical Methods, 2nd edn. Springer, New York (2004)

    MATH  Google Scholar 

  • Rubino, G., Tuffin, B. (eds.): Rare Event Simulation Using Monte Carlo Methods. Wiley, New York (2009)

    MATH  Google Scholar 

  • Rubinstein, R.Y., Kroese, D.P.: The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning. Springer, New York (2004)

    MATH  Google Scholar 

  • Rubinstein, R.Y., Kroese, D.P.: Simulation and the Monte Carlo Method, 2nd edn. Wiley, New York (2007)

    Book  Google Scholar 

  • Zhang, P.: Nonparametric importance sampling. J. Am. Stat. Assoc. 91(435), 1245–1253 (1996)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zdravko I. Botev.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Botev, Z.I., L’Ecuyer, P. & Tuffin, B. Markov chain importance sampling with applications to rare event probability estimation. Stat Comput 23, 271–285 (2013). https://doi.org/10.1007/s11222-011-9308-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-011-9308-2

Keywords

Navigation