The Annals of Statistics

Convergence of adaptive mixtures of importance sampling schemes

R. Douc, A. Guillin, J.-M. Marin, and C. P. Robert

Full-text: Open access

Enhanced PDF (859 KB)

Abstract

In the design of efficient simulation algorithms, one is often beset with a poor choice of proposal distributions. Although the performance of a given simulation kernel can clarify a posteriori how adequate this kernel is for the problem at hand, a permanent on-line modification of kernels causes concerns about the validity of the resulting algorithm. While the issue is most often intractable for MCMC algorithms, the equivalent version for importance sampling algorithms can be validated quite precisely. We derive sufficient convergence conditions for adaptive mixtures of population Monte Carlo algorithms and show that Rao–Blackwellized versions asymptotically achieve an optimum in terms of a Kullback divergence criterion, while more rudimentary versions do not benefit from repeated updating.

Article information

Source
Ann. Statist., Volume 35, Number 1 (2007), 420-448.

Dates
First available in Project Euclid: 6 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1181100193

Digital Object Identifier
doi:10.1214/009053606000001154

Mathematical Reviews number (MathSciNet)
MR2332281

Zentralblatt MATH identifier
1132.60022

Subjects
Primary: 60F05: Central limit and other weak theorems 62L12: Sequential estimation 65-04: Explicit machine computation and programs (not the theory of computation or programming) 65C05: Monte Carlo methods 65C40: Computational Markov chains 65C60: Computational problems in statistics

Keywords
Bayesian statistics Kullback divergence LLN MCMC algorithm population Monte Carlo proposal distribution Rao–Blackwellization

Citation

Douc, R.; Guillin, A.; Marin, J.-M.; Robert, C. P. Convergence of adaptive mixtures of importance sampling schemes. Ann. Statist. 35 (2007), no. 1, 420--448. doi:10.1214/009053606000001154. https://projecteuclid.org/euclid.aos/1181100193


Export citation

References

  • Agresti, A. (2002). Categorical Data Analysis, 2nd ed. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR1914507
    Zentralblatt MATH: 1018.62002
  • Andrieu, C. and Robert, C. (2001). Controlled Markov chain Monte Carlo methods for optimal sampling. Technical Report 0125, Univ. Paris Dauphine.
  • Cappé, O., Guillin, A., Marin, J. and Robert, C. (2004). Population Monte Carlo. J. Comput. Graph. Statist. 13 907–929.
    Mathematical Reviews (MathSciNet): MR2109057
    Digital Object Identifier: doi:10.1198/106186004X12803
  • Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2159833
  • Celeux, G., Marin, J. and Robert, C. (2006). Iterated importance sampling in missing data problems. Comput. Statist. Data Anal. 50 3386–3404.
    Mathematical Reviews (MathSciNet): MR2236856
    Zentralblatt MATH: 05381741
  • Chopin, N. (2004). Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 2385–2411.
    Mathematical Reviews (MathSciNet): MR2153989
    Digital Object Identifier: doi:10.1214/009053604000000698
    Project Euclid: euclid.aos/1107794873
    Zentralblatt MATH: 1079.65006
  • Csiszár, I. and Tusnády, G. (1984). Information geometry and alternating minimization procedures. Recent results in estimation theory and related topics. Statist. Decisions 1984 (suppl. 1) 205–237.
    Mathematical Reviews (MathSciNet): MR0785210
  • Del Moral, P., Doucet, A. and Jasra, A. (2006). Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 411–436.
    Mathematical Reviews (MathSciNet): MR2278333
    Digital Object Identifier: doi:10.1111/j.1467-9868.2006.00553.x
    Zentralblatt MATH: 1105.62034
  • Douc, R., Guillin, A., Marin, J. and Robert, C. (2005). Minimum variance importance sampling via population Monte Carlo. Technical report, Cahiers du CEREMADE, Univ. Paris Dauphine.
    Zentralblatt MATH: 1181.60028
    Digital Object Identifier: doi:10.1051/ps:2007028
  • Douc, R. and Moulines, E. (2005). Limit theorems for properly weighted samples with applications to sequential Monte Carlo. Technical report, TSI, Telecom Paris.
    Zentralblatt MATH: 1155.62056
    Digital Object Identifier: doi:10.1214/07-AOS514
    Project Euclid: euclid.aos/1223908095
  • Doucet, A., de Freitas, N. and Gordon, N., eds. (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1847783
    Zentralblatt MATH: 0967.00022
  • Gilks, W., Roberts, G. and Sahu, S. (1998). Adaptive Markov chain Monte Carlo through regeneration. J. Amer. Statist. Assoc. 93 1045–1054.
    Mathematical Reviews (MathSciNet): MR1649199
    Digital Object Identifier: doi:10.2307/2669848
    JSTOR: links.jstor.org
    Zentralblatt MATH: 1064.65503
  • Haario, H., Saksman, E. and Tamminen, J. (1999). Adaptive proposal distribution for random walk Metropolis algorithm. Comput. Statist. 14 375–395.
    Zentralblatt MATH: 0941.62036
    Digital Object Identifier: doi:10.1007/s001800050022
  • Haario, H., Saksman, E. and Tamminen, J. (2001). An adaptive Metropolis algorithm. Bernoulli 7 223–242.
    Mathematical Reviews (MathSciNet): MR1828504
    Digital Object Identifier: doi:10.2307/3318737
    Project Euclid: euclid.bj/1080222083
    Zentralblatt MATH: 0989.65004
  • Hesterberg, T. (1995). Weighted average importance sampling and defensive mixture distributions. Technometrics 37 185–194.
    Zentralblatt MATH: 0822.62002
    Digital Object Identifier: doi:10.1080/00401706.1995.10484303
  • Iba, Y. (2000). Population-based Monte Carlo algorithms. Trans. Japanese Society for Artificial Intelligence 16 279–286.
  • Künsch, H. (2005). Recursive Monte Carlo filters: Algorithms and theoretical analysis. Ann. Statist. 33 1983–2021.
    Mathematical Reviews (MathSciNet): MR2211077
    Digital Object Identifier: doi:10.1214/009053605000000426
    Project Euclid: euclid.aos/1132936554
    Zentralblatt MATH: 1086.62106
  • Mengersen, K. L. and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101–121.
    Mathematical Reviews (MathSciNet): MR1389882
    Digital Object Identifier: doi:10.1214/aos/1033066201
    Project Euclid: euclid.aos/1033066201
    Zentralblatt MATH: 0854.60065
  • Robert, C. (1996). Intrinsic losses. Theory and Decision 40 191–214.
    Mathematical Reviews (MathSciNet): MR1385186
    Digital Object Identifier: doi:10.1007/BF00133173
    Zentralblatt MATH: 0848.90010
  • Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2080278
    Zentralblatt MATH: 02117879
  • Roberts, G. O., Gelman, A. and Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 7 110–120.
    Mathematical Reviews (MathSciNet): MR1428751
    Digital Object Identifier: doi:10.1214/aoap/1034625254
    Project Euclid: euclid.aoap/1034625254
    Zentralblatt MATH: 0876.60015
  • Rubin, D. (1988). Using the SIR algorithm to simulate posterior distributions. In Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.) 395–402. Oxford Univ. Press.
    Zentralblatt MATH: 0713.62035
  • Sahu, S. and Zhigljavsky, A. (1998). Adaptation for self regenerative MCMC. Technical report, Univ. of Wales, Cardiff.
  • Sahu, S. and Zhigljavsky, A. (2003). Self regenerative Markov chain Monte Carlo with adaptation. Bernoulli 9 395–422.
    Mathematical Reviews (MathSciNet): MR1997490
    Digital Object Identifier: doi:10.3150/bj/1065444811
    Project Euclid: euclid.bj/1065444811
    Zentralblatt MATH: 1044.62033
  • Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701–1762.
    Mathematical Reviews (MathSciNet): MR1329166
    Digital Object Identifier: doi:10.1214/aos/1176325750
    Project Euclid: euclid.aos/1176325750
    Zentralblatt MATH: 0829.62080

The Institute of Mathematical Statistics

The Annals of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more