Statistics and Computing

, Volume 22, Issue 6, pp 1167–1180 | Cite as

Approximate Bayesian computational methods

  • Jean-Michel Marin
  • Pierre Pudlo
  • Christian P. Robert
  • Robin J. Ryder


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.


Likelihood-free methods Bayesian statistics ABC methodology DIYABC Bayesian model choice 


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  1. Bartolucci, F., Scaccia, L., Mira, A.: Efficient Bayes factor estimation from the reversible jump output. Biometrika 93(1), 41–52 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  2. Beaumont, M., Zhang, W., Balding, D.: Approximate Bayesian computation in population genetics. Genetics 162(4), 2025–2035 (2002) Google Scholar
  3. Beaumont, M., Cornuet, J.-M., Marin, J.-M., Robert, C.: Adaptive approximate Bayesian computation. Biometrika 96(4), 983–990 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 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) CrossRefGoogle Scholar
  5. Berger, J., Fienberg, S., Raftery, A., Robert, C.: Incoherent phylogeographic inference. Proc. Natl. Acad. Sci. 107(41), E57 (2010) CrossRefGoogle Scholar
  6. Blum, M.: Approximate Bayesian computation: a non-parametric perspective. J. Am. Stat. Assoc. 105(491), 1178–1187 (2010) CrossRefGoogle Scholar
  7. Blum, M., François, O.: Non-linear regression models for approximate Bayesian computation. Stat. Comput. 20(1), 63–73 (2010) MathSciNetCrossRefGoogle Scholar
  8. Calvet, L., Czellar, V.: State-observation sampling and the econometrics of learning models. Technical Report (2011). arXiv:1105.4519
  9. 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) CrossRefGoogle Scholar
  10. 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) CrossRefGoogle Scholar
  11. Csillèry, K., Blum, M., Gaggiotti, O., François, O.: Approximate Bayesian computation (ABC) in practice. Trends Ecol. Evol. 25(7), 410–418 (2010a) CrossRefGoogle Scholar
  12. 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) CrossRefGoogle Scholar
  13. 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) MathSciNetCrossRefGoogle Scholar
  14. 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
  15. Del Moral, P., Doucet, A., Jasra, A.: Sequential Monte Carlo samplers. J. R. Stat. Soc. B 68(3), 411–436 (2006) zbMATHCrossRefGoogle Scholar
  16. Del Moral, P., Doucet, A., Jasra, A.: An adaptive sequential Monte Carlo method for approximate Bayesian computation. Stat. Comput. (2011, to appear) Google Scholar
  17. Didelot, X., Everitt, R., Johansen, A., Lawson, D.: Likelihood-free estimation of model evidence. Bayesian Anal. 6(1), 48–76 (2011) MathSciNetCrossRefGoogle Scholar
  18. Douc, R., Guillin, A., Marin, J.-M., Robert, C.: Convergence of adaptive mixtures of importance sampling schemes. Ann. Stat. 35(1), 420–448 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  19. Drovandi, C., Pettitt, A.: Estimation of parameters for macroparasite population evolution using approximate Bayesian computation. Biometrics 67(1), 225–233 (2010) MathSciNetCrossRefGoogle Scholar
  20. Fearnhead, P., Prangle, D.: Semi-automatic approximate Bayesian computation. Technical Report (2010). arXiv:1004.1112
  21. Friel, N., Pettitt, A.: Marginal likelihood estimation via power posteriors. J. R. Stat. Soc. B 70(3), 589–607 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  22. Gauchi, J.-P., Vila, J.-P.: Nonparametric filtering approaches for identification and inference in nonlinear dynamic systems. Technical report. Personal communication (2011) Google Scholar
  23. Gelfand, A., Smith, A.: Sampling based approaches to calculating marginal densities. J. Am. Stat. Assoc. 85(410), 398–409 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  24. Gouriéroux, C., Monfort, A., Renault, E.: Indirect inference. J. Appl. Econom. 8, 85–118 (1993) CrossRefGoogle Scholar
  25. 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
  26. Jaakkola, T., Jordan, M.: Bayesian parameter estimation via variational methods. Stat. Comput. 10(1), 25–37 (2000) CrossRefGoogle Scholar
  27. Jasra, A., Singh, S.S., Martin, J.S., McCoy, E.: Filtering via approximate Bayesian computation. Stat. Comput. (2011, to appear) Google Scholar
  28. Joyce, P., Marjoram, P.: Approximately sufficient statistics and Bayesian computation. Stat. Appl. Genet. Mol. Biol. 7(1), 26 (2008) MathSciNetGoogle Scholar
  29. Leuenberger, C., Wegmann, D., Excoffier, L.: Bayesian computation and model selection in population genetics. Genetics 184(1), 243–252 (2010) CrossRefGoogle Scholar
  30. Marin, J.-M., Robert, C.: Bayesian Core. Springer, New York (2007) zbMATHGoogle Scholar
  31. 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
  32. Marjoram, P., Molitor, J., Plagnol, V., Tavaré, S.: Markov chain Monte Carlo without likelihoods. Proc. Natl. Acad. Sci. 100(26), 15324–15328 (2003) CrossRefGoogle Scholar
  33. McKinley, T., Cook, A., Deardon, R.: Inference in epidemic models without likelihoods. Int. J. Biostat. 5(1), 24 (2009) MathSciNetGoogle Scholar
  34. 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) MathSciNetCrossRefGoogle Scholar
  35. 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) CrossRefGoogle Scholar
  36. R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2006) Google Scholar
  37. Ratmann, O.: ABC under model uncertainty. PhD thesis, Imperial College, London (2009) Google Scholar
  38. 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
  39. 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) CrossRefGoogle Scholar
  40. Robert, C.: The Bayesian Choice, 2nd edn. Springer, New York (2001) zbMATHGoogle Scholar
  41. Robert, C., Casella, G.: Monte Carlo Statistical Methods, 2nd edn. Springer, New York (2004) zbMATHGoogle Scholar
  42. Robert, C.P., Mengersen, K., Chen, C.: Model choice versus model criticism. Proc. Natl. Acad. Sci. 107(3), E5 (2010) CrossRefGoogle Scholar
  43. 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) CrossRefGoogle Scholar
  44. Rubin, D.: Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Stat. 12(4), 1151–1172 (1984) zbMATHCrossRefGoogle Scholar
  45. Rue, H., Held, L.: Gaussian Markov Random Fields: Theory and Applications. Monographs on Statistics and Applied Probability, vol. 104. Chapman & Hall, London (2005) zbMATHCrossRefGoogle Scholar
  46. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  47. Sisson, S.A., Fan, Y., Tanaka, M.: Sequential Monte Carlo without likelihoods. Proc. Natl. Acad. Sci. 104(6), 1760–1765 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  48. Sisson, S.A., Fan, Y., Tanaka, M.: Sequential Monte Carlo without likelihoods: Errata. Proc. Natl. Acad. Sci. 106(39), 16889 (2009) Google Scholar
  49. Tavaré, S., Balding, D., Griffith, R., Donnelly, P.: Inferring coalescence times from DNA sequence data. Genetics 145(2), 505–518 (1997) Google Scholar
  50. Templeton, A.: Statistical hypothesis testing in intraspecific phylogeography: nested clade phylogeographical analysis vs. approximate Bayesian computation. Mol. Ecol. 18(2), 319–331 (2008) MathSciNetCrossRefGoogle Scholar
  51. Templeton, A.: Coherent and incoherent inference in phylogeography and human evolution. Proc. Natl. Acad. Sci. 107(14), 6376–6381 (2010) CrossRefGoogle Scholar
  52. Tierney, L., Kadane, J.: Accurate approximations for posterior moments and marginal densities. J. Am. Stat. Assoc. 81(393), 82–86 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  53. Toni, T., Stumpf, M.: Simulation-based model selection for dynamical systems in systems and population biology. Bioinformatics 26(1), 104–110 (2010) CrossRefGoogle Scholar
  54. 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) CrossRefGoogle Scholar
  55. Wasserman, L.: All of Nonparametric Statistics. Springer, New York (2007) Google Scholar
  56. Wilkinson, R.D.: Approximate Bayesian computation (ABC) gives exact results under the assumption of model error. Technical Report (2008). arXiv:0811.3355

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Jean-Michel Marin
    • 1
  • Pierre Pudlo
    • 1
  • Christian P. Robert
    • 2
  • Robin J. Ryder
    • 2
  1. 1.I3M, UMR CNRS 5149Université Montpellier 2MontpellierFrance
  2. 2.CEREMADEUniversité Paris Dauphine and Crest INSEEParisFrance

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