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Approximate Bayesian computational methods


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.

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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)

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

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

  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)

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

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

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

  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)

  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)

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

  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)

  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)

  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)

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

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

  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)

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

  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)

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

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

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

  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)

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

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

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

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

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

  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)

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

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

  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)

  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)

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

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

  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)

  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)

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

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

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

  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)

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

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

  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)

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

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

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

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

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

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

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

  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)

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

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

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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.

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Marin, J., 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

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