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Asymptotic Properties of Approximate Bayesian Computation

Author

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  • D.T. Frazier
  • G.M. Martin
  • C.P. Robert
  • J. Rousseau

Abstract

Approximate Bayesian computation (ABC) is becoming an accepted tool for statistical analysis in models with intractable likelihoods. With the initial focus being primarily on the practical import of ABC, exploration of its formal statistical properties has begun to attract more attention. In this paper we consider the asymptotic behaviour of the posterior obtained from ABC and the ensuing posterior mean. We give general results on: (i) the rate of concentration of the ABC posterior on sets containing the true parameter (vector); (ii) the limiting shape of the posterior; and (iii) the asymptotic distribution of the ABC posterior mean. These results hold under given rates for the tolerance used within ABC, mild regularity conditions on the summary statistics, and a condition linked to identification of the true parameters. Important implication of the theoretical results for practitioners of ABC are highlighted.

Suggested Citation

  • D.T. Frazier & G.M. Martin & C.P. Robert & J. Rousseau, 2016. "Asymptotic Properties of Approximate Bayesian Computation," Monash Econometrics and Business Statistics Working Papers 18/16, Monash University, Department of Econometrics and Business Statistics.
    • Handle: RePEc:msh:ebswps:2016-18
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    File URL: http://business.monash.edu/econometrics-and-business-statistics/research/publications/ebs/wp18-16.pdf
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    References listed on IDEAS

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    1. repec:dau:papers:123456789/7848 is not listed on IDEAS
    2. Paul Fearnhead & Dennis Prangle, 2012. "Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(3), pages 419-474, June.
    3. Christopher C. Drovandi & Anthony N. Pettitt & Malcolm J. Faddy, 2011. "Approximate Bayesian computation using indirect inference," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 60(3), pages 317-337, May.
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    5. Joyce Paul & Marjoram Paul, 2008. "Approximately Sufficient Statistics and Bayesian Computation," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 7(1), pages 1-18, August.
    6. Jean-Michel Marin & Natesh S. Pillai & Christian P. Robert & Judith Rousseau, 2014. "Relevant statistics for Bayesian model choice," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(5), pages 833-859, November.
    7. Knut Heggland & Arnoldo Frigessi, 2004. "Estimating functions in indirect inference," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(2), pages 447-462, May.
    8. Blum, Michael G. B., 2010. "Approximate Bayesian Computation: A Nonparametric Perspective," Journal of the American Statistical Association, American Statistical Association, vol. 105(491), pages 1178-1187.
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    Citations

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    Cited by:

    1. Frazier, David T. & Maneesoonthorn, Worapree & Martin, Gael M. & McCabe, Brendan P.M., 2019. "Approximate Bayesian forecasting," International Journal of Forecasting, Elsevier, vol. 35(2), pages 521-539.
    2. David T. Frazier & Christian P. Robert & Judith Rousseau, 2017. "Model Misspecification in ABC: Consequences and Diagnostics," Papers 1708.01974, arXiv.org, revised Jul 2019.

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    More about this item

    Keywords

    asymptotic properties; Bayesian inference; Bernstein-von Mises theorem; consistency; likelihood-free methods;
    All these keywords.

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General

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