Skip to main content
SpringerLink
Log in

Approximate Bayesian computational methods

Statistics and Computing Aims and scope Submit manuscript

Cite this article

Abstract

Approximate Bayesian Computation (ABC) methods, also known as likelihood-free techniques, have appeared in the past ten years as the most satisfactory approach to intractable likelihood problems, first in genetics then in a broader spectrum of applications. However, these methods suffer to some degree from calibration difficulties that make them rather volatile in their implementation and thus render them suspicious to the users of more traditional Monte Carlo methods. In this survey, we study the various improvements and extensions brought on the original ABC algorithm in recent years.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

References

  • Bartolucci, F., Scaccia, L., Mira, A.: Efficient Bayes factor estimation from the reversible jump output. Biometrika 93(1), 41–52 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Beaumont, M., Zhang, W., Balding, D.: Approximate Bayesian computation in population genetics. Genetics 162(4), 2025–2035 (2002)

    Google Scholar 

  • Beaumont, M., Cornuet, J.-M., Marin, J.-M., Robert, C.: Adaptive approximate Bayesian computation. Biometrika 96(4), 983–990 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Beaumont, M., Nielsen, R., Robert, C., Hey, J., Gaggiotti, O., Knowles, L., Estoup, A., Mahesh, P., Coranders, J., Hickerson, M., Sisson, S., Fagundes, N., Chikhi, L., Beerli, P., Vitalis, R., Cornuet, J.-M., Huelsenbeck, J., Foll, M., Yang, Z., Rousset, F., Balding, D., Excoffier, L.: In defense of model-based inference in phylogeography. Mol. Ecol. 19(3), 436–446 (2010)

    Article  Google Scholar 

  • Berger, J., Fienberg, S., Raftery, A., Robert, C.: Incoherent phylogeographic inference. Proc. Natl. Acad. Sci. 107(41), E57 (2010)

    Article  Google Scholar 

  • Blum, M.: Approximate Bayesian computation: a non-parametric perspective. J. Am. Stat. Assoc. 105(491), 1178–1187 (2010)

    Article  Google Scholar 

  • Blum, M., François, O.: Non-linear regression models for approximate Bayesian computation. Stat. Comput. 20(1), 63–73 (2010)

    Article  MathSciNet  Google Scholar 

  • Calvet, L., Czellar, V.: State-observation sampling and the econometrics of learning models. Technical Report (2011). arXiv:1105.4519

  • Campillo, F., Rossi, V.: Convolution particle filter for parameter estimation in general state-space models. IEEE Trans. Aerosp. Electron. Syst. 45(3), 1063–1072 (2009)

    Article  Google Scholar 

  • Cornuet, J.-M., Santos, F., Beaumont, M.A., Robert, C.P., Marin, J.-M., Balding, D.J., Guillemaud, T., Estoup, A.: Inferring population history with DIYABC: a user-friendly approach to approximate Bayesian computation. Bioinformatics 24(23), 2713–2719 (2008)

    Article  Google Scholar 

  • Csillèry, K., Blum, M., Gaggiotti, O., François, O.: Approximate Bayesian computation (ABC) in practice. Trends Ecol. Evol. 25(7), 410–418 (2010a)

    Article  Google Scholar 

  • Csillèry, K., Blum, M., Gaggiotti, O., François, O.: Invalid arguments against ABC: a reply to A.R. Templeton. Trends Ecol. Evol. 25(7), 490–491 (2010b)

    Article  Google Scholar 

  • Cucala, L., Marin, J.-M., Robert, C., Titterington, D.: Bayesian inference in k-nearest-neighbour classification models. J. Am. Stat. Assoc. 104(485), 263–273 (2009)

    Article  MathSciNet  Google Scholar 

  • Dean, T.A., Singh, S.S., Jasra, A., Peters, G.W.: Parameter estimation for hidden Markov models with intractable likelihoods. Technical Report (2011). arXiv:1103.5399

  • Del Moral, P., Doucet, A., Jasra, A.: Sequential Monte Carlo samplers. J. R. Stat. Soc. B 68(3), 411–436 (2006)

    Article  MATH  Google Scholar 

  • Del Moral, P., Doucet, A., Jasra, A.: An adaptive sequential Monte Carlo method for approximate Bayesian computation. Stat. Comput. (2011, to appear)

  • Didelot, X., Everitt, R., Johansen, A., Lawson, D.: Likelihood-free estimation of model evidence. Bayesian Anal. 6(1), 48–76 (2011)

    Article  MathSciNet  Google Scholar 

  • Douc, R., Guillin, A., Marin, J.-M., Robert, C.: Convergence of adaptive mixtures of importance sampling schemes. Ann. Stat. 35(1), 420–448 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Drovandi, C., Pettitt, A.: Estimation of parameters for macroparasite population evolution using approximate Bayesian computation. Biometrics 67(1), 225–233 (2010)

    Article  MathSciNet  Google Scholar 

  • Fearnhead, P., Prangle, D.: Semi-automatic approximate Bayesian computation. Technical Report (2010). arXiv:1004.1112

  • Friel, N., Pettitt, A.: Marginal likelihood estimation via power posteriors. J. R. Stat. Soc. B 70(3), 589–607 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Gauchi, J.-P., Vila, J.-P.: Nonparametric filtering approaches for identification and inference in nonlinear dynamic systems. Technical report. Personal communication (2011)

  • Gelfand, A., Smith, A.: Sampling based approaches to calculating marginal densities. J. Am. Stat. Assoc. 85(410), 398–409 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Gouriéroux, C., Monfort, A., Renault, E.: Indirect inference. J. Appl. Econom. 8, 85–118 (1993)

    Article  Google Scholar 

  • Grelaud, A., Marin, J.-M., Robert, C., Rodolphe, F., Tally, F.: Likelihood-free methods for model choice in Gibbs random fields. Bayesian Anal. 3(2), 427–442 (2009)

    Google Scholar 

  • Jaakkola, T., Jordan, M.: Bayesian parameter estimation via variational methods. Stat. Comput. 10(1), 25–37 (2000)

    Article  Google Scholar 

  • Jasra, A., Singh, S.S., Martin, J.S., McCoy, E.: Filtering via approximate Bayesian computation. Stat. Comput. (2011, to appear)

  • Joyce, P., Marjoram, P.: Approximately sufficient statistics and Bayesian computation. Stat. Appl. Genet. Mol. Biol. 7(1), 26 (2008)

    MathSciNet  Google Scholar 

  • Leuenberger, C., Wegmann, D., Excoffier, L.: Bayesian computation and model selection in population genetics. Genetics 184(1), 243–252 (2010)

    Article  Google Scholar 

  • Marin, J.-M., Robert, C.: Bayesian Core. Springer, New York (2007)

    MATH  Google Scholar 

  • Marin, J.-M., Robert, C.: Importance sampling methods for Bayesian discrimination between embedded models. In: Chen, M.-H., Dey, D., Müller, P., Sun, D., Ye, K. (eds.) Frontiers of Statistical Decision Making and Bayesian Analysis, pp. 513–527. Springer, New York (2010)

    Google Scholar 

  • Marjoram, P., Molitor, J., Plagnol, V., Tavaré, S.: Markov chain Monte Carlo without likelihoods. Proc. Natl. Acad. Sci. 100(26), 15324–15328 (2003)

    Article  Google Scholar 

  • McKinley, T., Cook, A., Deardon, R.: Inference in epidemic models without likelihoods. Int. J. Biostat. 5(1), 24 (2009)

    MathSciNet  Google Scholar 

  • Møller, J., Pettitt, A., Reeves, R., Berthelsen, K.: An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika 93(2), 451–458 (2006)

    Article  MathSciNet  Google Scholar 

  • Pritchard, J., Seielstad, M., Perez-Lezaun, A., Feldman, M.: Population growth of human Y chromosomes: a study of Y chromosome microsatellites. Mol. Biol. Evol. 16(12), 1791–1798 (1999)

    Article  Google Scholar 

  • R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2006)

    Google Scholar 

  • Ratmann, O.: ABC under model uncertainty. PhD thesis, Imperial College, London (2009)

  • Ratmann, O., Andrieu, C., Wiujf, C., Richardson, S.: Model criticism based on likelihood-free inference, with an application to protein network evolution. Proc. Natl. Acad. Sci. 106(26), 1–6 (2009)

    Google Scholar 

  • Ratmann, O., Andrieu, C., Wiuf, C., Richardson, S.: Reply to Robert et al.: Model criticism informs model choice and model comparison. Proc. Natl. Acad. Sci. 107(3), E6 (2010)

    Article  Google Scholar 

  • Robert, C.: The Bayesian Choice, 2nd edn. Springer, New York (2001)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Robert, C.P., Mengersen, K., Chen, C.: Model choice versus model criticism. Proc. Natl. Acad. Sci. 107(3), E5 (2010)

    Article  Google Scholar 

  • Robert, C.P., Cornuet, J.-M., Marin, J.-M., Pillai, N.: Lack of confidence in ABC model choice. Proc. Natl. Acad. Sci. 108(37), 15112–15117 (2011)

    Article  Google Scholar 

  • Rubin, D.: Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Stat. 12(4), 1151–1172 (1984)

    Article  MATH  Google Scholar 

  • Rue, H., Held, L.: Gaussian Markov Random Fields: Theory and Applications. Monographs on Statistics and Applied Probability, vol. 104. Chapman & Hall, London (2005)

    Book  MATH  Google Scholar 

  • Rue, H., Martino, S., Chopin, N.: Approximate Bayesian inference for latent Gaussian models using integrated nested Laplace approximations. J. R. Stat. Soc. B 71(2), 319–392 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Sisson, S.A., Fan, Y., Tanaka, M.: Sequential Monte Carlo without likelihoods. Proc. Natl. Acad. Sci. 104(6), 1760–1765 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Sisson, S.A., Fan, Y., Tanaka, M.: Sequential Monte Carlo without likelihoods: Errata. Proc. Natl. Acad. Sci. 106(39), 16889 (2009)

    Google Scholar 

  • Tavaré, S., Balding, D., Griffith, R., Donnelly, P.: Inferring coalescence times from DNA sequence data. Genetics 145(2), 505–518 (1997)

    Google Scholar 

  • Templeton, A.: Statistical hypothesis testing in intraspecific phylogeography: nested clade phylogeographical analysis vs. approximate Bayesian computation. Mol. Ecol. 18(2), 319–331 (2008)

    Article  MathSciNet  Google Scholar 

  • Templeton, A.: Coherent and incoherent inference in phylogeography and human evolution. Proc. Natl. Acad. Sci. 107(14), 6376–6381 (2010)

    Article  Google Scholar 

  • Tierney, L., Kadane, J.: Accurate approximations for posterior moments and marginal densities. J. Am. Stat. Assoc. 81(393), 82–86 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  • Toni, T., Stumpf, M.: Simulation-based model selection for dynamical systems in systems and population biology. Bioinformatics 26(1), 104–110 (2010)

    Article  Google Scholar 

  • Toni, T., Welch, D., Strelkowa, N., Ipsen, A., Stumpf, M.: Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J. R. Soc. Interface 6(31), 187–202 (2009)

    Article  Google Scholar 

  • Wasserman, L.: All of Nonparametric Statistics. Springer, New York (2007)

    Google Scholar 

  • Wilkinson, R.D.: Approximate Bayesian computation (ABC) gives exact results under the assumption of model error. Technical Report (2008). arXiv:0811.3355

Download references

Author information

Authors and Affiliations

  1. I3M, UMR CNRS 5149, Université Montpellier 2, Montpellier, France

    Jean-Michel Marin & Pierre Pudlo

  2. CEREMADE, Université Paris Dauphine and Crest INSEE, Paris, France

    Christian P. Robert & Robin J. Ryder

Authors
  1. Jean-Michel Marin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre Pudlo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian P. Robert
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Robin J. Ryder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Michel Marin.

Additional information

This research was financially supported by the French Agence Nationale de la Recherche grant ‘EMILE’ ANR-09-BLAN-0145-01, as well as by the Fondation des Sciences Mathématiques de Paris and a GIS scholarship for the fourth author.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Marin, JM., Pudlo, P., Robert, C.P. et al. Approximate Bayesian computational methods. Stat Comput 22, 1167–1180 (2012). https://doi.org/10.1007/s11222-011-9288-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-011-9288-2

Keywords

search

Navigation