• Bernoulli
  • Volume 25, Number 3 (2019), 1977-1998.

Consistency of adaptive importance sampling and recycling schemes

Jean-Michel Marin, Pierre Pudlo, and Mohammed Sedki

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Among Monte Carlo techniques, the importance sampling requires fine tuning of a proposal distribution, which is now fluently resolved through iterative schemes. Sequential adaptive algorithms have been proposed to calibrate the sampling distribution. Cornuet et al. [Scand. J. Stat. 39 (2012) 798–812] provides a significant improvement in stability and effective sample size by the introduction of a recycling procedure. However, the consistency of such algorithms have been rarely tackled because of their complexity. Moreover, the recycling strategy of the AMIS estimator adds another difficulty and its consistency remains largely open. In this work, we prove the convergence of sequential adaptive sampling, with finite Monte Carlo sample size at each iteration, and consistency of recycling procedures. Contrary to Douc et al. [Ann. Statist. 35 (2007) 420–448], results are obtained here in the asymptotic regime where the number of iterations is going to infinity while the number of drawings per iteration is a fixed, but growing sequence of integers. Hence, some of the results shed new light on adaptive population Monte Carlo algorithms in that last regime and give advices on how the sample sizes should be fixed.

Article information

Bernoulli, Volume 25, Number 3 (2019), 1977-1998.

Received: March 2017
First available in Project Euclid: 12 June 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

adaptive algorithms importance sampling Monte Carlo methods population Monte Carlo sequential Monte Carlo triangular arrays


Marin, Jean-Michel; Pudlo, Pierre; Sedki, Mohammed. Consistency of adaptive importance sampling and recycling schemes. Bernoulli 25 (2019), no. 3, 1977--1998. doi:10.3150/18-BEJ1042.

Export citation


  • [1] Andrieu, C., Doucet, A. and Holenstein, R. (2010). Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 269–342.
  • [2] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. New York: Wiley.
  • [3] Bugallo, M.F., Martino, L. and Corander, J. (2015). Adaptive importance sampling in signal processing. Digit. Signal Process. 47 36–49.
  • [4] Cameron, E. and Pettitt, A. (2014). Recursive pathways to marginal likelihood estimation with prior-sensitivity analysis. Statist. Sci. 29 397–419.
  • [5] Cappé, O., Guillin, A., Marin, J.M. and Robert, C.P. (2004). Population Monte Carlo. J. Comput. Graph. Statist. 13 907–929.
  • [6] Cappé, O., Guillin, A., Marin, J.-M. and Robert, C.P. (2008). Adaptive importance sampling in general mixture classes. Stat. Comput. 18 587–600.
  • [7] Cornuet, J.-M., Marin, J.-M., Mira, A. and Robert, C.P. (2012). Adaptive multiple importance sampling. Scand. J. Stat. 39 798–812.
  • [8] Del Moral, P., Doucet, A. and Jasra, A. (2006). Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 411–436.
  • [9] Douc, R., Guillin, A., Marin, J.-M. and Robert, C.P. (2007). Convergence of adaptive mixtures of importance sampling schemes. Ann. Statist. 35 420–448.
  • [10] Douc, R., Guillin, A., Marin, J.-M. and Robert, C.P. (2007). Minimum variance importance sampling via population Monte Carlo. ESAIM Probab. Stat. 11 427–447.
  • [11] Feroz, F., Hobson, M., Cameron, E. and Pettitt, A. (2013). Importance nested sampling and the MultiNest algorithm. Preprint. Available at arXiv:1306.2144.
  • [12] Forbes, F. and Fort, G. (2007). Combining Monte Carlo and mean-field-like methods for inference in hidden Markov random fields. IEEE Trans. Image Process. 16 824–837.
  • [13] He, H.Y. and Owen, A.B. (2014). Optimal mixture weights in multiple importance sampling. Preprint. Available at arXiv:1411.3954.
  • [14] Hesterberg, T. (1988). Advances in importance sampling. Ph.D. thesis, Stanford University.
  • [15] Hesterberg, T. (1995). Weighted average importance sampling and defensive mixture distributions. Technometrics 37 185–194.
  • [16] Liu, J.S. (2008). Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics. New York: Springer.
  • [17] Martino, L., Elvira, V., Luengo, D. and Corander, J. (2015). An adaptive population importance sampler: Learning from uncertainty. IEEE Trans. Signal Process. 63 4422–4437.
  • [18] Martino, L., Elvira, V., Luengo, D. and Corander, J. (2017). Layered adaptive importance sampling. Stat. Comput. 27 599–623.
  • [19] McLachlan, G.J. and Krishnan, T. (2007). The EM Algorithm and Extensions. New York: Wiley.
  • [20] Owen, A. and Zhou, Y. (2000). Safe and effective importance sampling. J. Amer. Statist. Assoc. 95 135–143.
  • [21] Ripley, B.D. (1987). Stochastic Simulation. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. New York: Wiley.
  • [22] Robert, C.P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer Texts in Statistics. New York: Springer.
  • [23] Schuster, I. (2015). Gradient importance sampling. Preprint. Available at arXiv:1507.05781.
  • [24] Schuster, I. (2015). Consistency of importance sampling estimates based on dependent sample sets and an application to models with factorizing likelihoods. Preprint. Available at arXiv:1503.00357.
  • [25] Sirén, J., Marttinen, P. and Corander, J. (2010). Reconstructing population histories from single-nucleotide polymorphism data. Mol. Biol. Evol. 28 673–683.
  • [26] Šmídl, V. and Hofman, R. (2014). Efficient sequential Monte Carlo sampling for continuous monitoring of a radiation situation. Technometrics 56 514–528.
  • [27] Van der Vaart, A.W. (2000). Asymptotic Statistics. Cambridge: Cambridge University Press.
  • [28] Veach, E. and Guibas, L.J. (1995). Optimally comabining sampling techniques for Monte Carlo rendering. In SIGGRAPH’95 Proceeding 419–428. Addison-Wesley.
  • [29] Xiong, X., Šmídl, V. and Filippone, M. (2017). Adaptive multiple importance sampling for Gaussian processes. J. Stat. Comput. Simul. 87 1644–1665.