Asymptotic properties of approximate Bayesian computation

Summary Approximate Bayesian computation allows for statistical analysis using models with intractable likelihoods. In this paper we consider the asymptotic behaviour of the posterior distribution obtained by this method. We give general results on the rate at which the posterior distribution concentrates on sets containing the true parameter, the limiting shape of the posterior distribution, and the asymptotic distribution of the posterior mean. These results hold under given rates for the tolerance used within the method, mild regularity conditions on the summary statistics, and a condition linked to identification of the true parameters. Implications for practitioners are discussed.

[1]  M. Gutmann,et al.  Approximate Bayesian Computation , 2019, Annual Review of Statistics and Its Application.

[2]  Paul Fearnhead,et al.  Convergence of regression‐adjusted approximate Bayesian computation , 2016, 1609.07135.

[3]  David T. Frazier,et al.  Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models , 2016, Journal of Computational and Graphical Statistics.

[4]  Han Hong,et al.  Bayesian Indirect Inference and the ABC of GMM , 2015, 1512.07385.

[5]  Paul Fearnhead,et al.  On the Asymptotic Efficiency of Approximate Bayesian Computation Estimators , 2015, 1506.03481.

[6]  Dennis Kristensen,et al.  ABC of SV: Limited Information Likelihood Inference in Stochastic Volatility Jump-Diffusion Models , 2015 .

[7]  Anthony N. Pettitt,et al.  Bayesian indirect inference using a parametric auxiliary model , 2015, 1505.03372.

[8]  Ajay Jasra,et al.  Approximate Bayesian Computation for a Class of Time Series Models , 2014, 1401.0265.

[9]  Gilles Celeux,et al.  Approximate Bayesian computation methods , 2012, Statistics and Computing.

[10]  Arnaud Guyader,et al.  New insights into Approximate Bayesian Computation , 2012, 1207.6461.

[11]  Paul Fearnhead,et al.  Constructing summary statistics for approximate Bayesian computation: semi‐automatic approximate Bayesian computation , 2012 .

[12]  S. Sisson,et al.  A comparative review of dimension reduction methods in approximate Bayesian computation , 2012, 1202.3819.

[13]  D. J. Nott,et al.  Approximate Bayesian Computation and Bayes’ Linear Analysis: Toward High-Dimensional ABC , 2011, 1112.4755.

[14]  J.-M. Marin,et al.  Relevant statistics for Bayesian model choice , 2011, 1110.4700.

[15]  A. Pettitt,et al.  Approximate Bayesian computation using indirect inference , 2011 .

[16]  S. Sisson,et al.  Likelihood-free Markov chain Monte Carlo , 2010, 1001.2058.

[17]  L. Excoffier,et al.  Efficient Approximate Bayesian Computation Coupled With Markov Chain Monte Carlo Without Likelihood , 2009, Genetics.

[18]  M. Blum.  Approximate Bayesian Computation: A Nonparametric Perspective , 2009, 0904.0635.

[19]  Olivier François,et al.  Non-linear regression models for Approximate Bayesian Computation , 2008, Stat. Comput..

[20]  C. Robert,et al.  Adaptive approximate Bayesian computation , 2008, 0805.2256.

[21]  Mark M. Tanaka,et al.  Sequential Monte Carlo without likelihoods , 2007, Proceedings of the National Academy of Sciences.

[22]  K. Heggland,et al.  Estimating functions in indirect inference , 2004 .

[23]  Paul Marjoram,et al.  Markov chain Monte Carlo without likelihoods , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[24]  D. Balding,et al.  Approximate Bayesian computation in population genetics. , 2002, Genetics.

[25]  M. Feldman,et al.  Population growth of human Y chromosomes: a study of Y chromosome microsatellites. , 1999, Molecular biology and evolution.

[26]  P. Donnelly,et al.  Inferring coalescence times from DNA sequence data. , 1997, Genetics.

[27]  A. Gallant,et al.  Which Moments to Match? , 1995, Econometric Theory.

[28]  D. Duffie,et al.  Simulated Moments Estimation of Markov Models of Asset Prices , 1990 .

[29]  Chan‐Fu Chen.  On Asymptotic Normality of Limiting Density Functions with Bayesian Implications , 1985 .

[30]  R. Rao,et al.  Normal Approximation and Asymptotic Expansions , 1976 .

[31]  Paul Marjoram,et al.  Statistical Applications in Genetics and Molecular Biology Approximately Sufficient Statistics and Bayesian Computation , 2011 .

[32]  Christian Gourieroux,et al.  Simulation-based econometric methods , 1996 .

[33]  T. Amemiya.  Non-linear regression models , 1983 .

[34]  R. Z. Khasʹminskiĭ,et al.  Statistical estimation : asymptotic theory , 1981 .

[35]  A. M. Walker.  On the Asymptotic Behaviour of Posterior Distributions , 1969 .

[36]  Le Cam,et al.  On some asymptotic properties of maximum likelihood estimates and related Bayes' estimates , 1953 .