Convergence of regression‐adjusted approximate Bayesian computation

&NA; We present asymptotic results for the regression‐adjusted version of approximate Bayesian computation introduced by Beaumont et al. (2002). We show that for an appropriate choice of the bandwidth, regression adjustment will lead to a posterior that, asymptotically, correctly quantifies uncertainty. Furthermore, for such a choice of bandwidth we can implement an importance sampling algorithm to sample from the posterior whose acceptance probability tends to unity as the data sample size increases. This compares favourably to results for standard approximate Bayesian computation, where the only way to obtain a posterior that correctly quantifies uncertainty is to choose a much smaller bandwidth, one for which the acceptance probability tends to zero and hence for which Monte Carlo error will dominate.

[1]  Jean-Marie Cornuet,et al.  Inferring population history with DIY ABC: a user-friendly approach to approximate Bayesian computation , 2008, Bioinform..

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

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

[4]  Guillaume Deffuant,et al.  Adaptive approximate Bayesian computation for complex models , 2011, Computational Statistics.

[5]  J. Møller.  Discussion on the paper by Feranhead and Prangle , 2012 .

[6]  J. Ghosh,et al.  On convergence of posterior distributions , 1995 .

[7]  Jayanta K. Ghosh,et al.  Stability and Convergence of the Posterior in Non-Regular Problems , 1994 .

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

[9]  David Wooff,et al.  Bayes Linear Statistics: Theory and Methods , 2007 .

[10]  R. Plevin,et al.  Approximate Bayesian Computation in Evolution and Ecology , 2011 .

[11]  Laura Ventura,et al.  Bayesian composite marginal likelihoods , 2011 .

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

[13]  David Welch,et al.  Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems , 2009, Journal of The Royal Society Interface.

[14]  Julien Cornebise,et al.  On optimality of kernels for approximate Bayesian computation using sequential Monte Carlo , 2011, Statistical applications in genetics and molecular biology.

[15]  S. Soubeyrand,et al.  Weak convergence of posteriors conditional on maximum pseudo-likelihood estimates and implications in ABC , 2015 .

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

[17]  Carolyn Pillers Dobler.  Mathematical Statistics: Basic Ideas and Selected Topics , 2002 .

[18]  Laurent E. Calvet,et al.  Accurate Methods for Approximate Bayesian Computation Filtering , 2015 .

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

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

[21]  A. Davison,et al.  Bayesian Inference from Composite Likelihoods, with an Application to Spatial Extremes , 2009, 0911.5357.

[22]  Erlis Ruli,et al.  Approximate Bayesian computation with composite score functions , 2013, Stat. Comput..

[23]  M. Gutmann,et al.  Approximate Bayesian Computation , 2012 .

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

[25]  Paul Fearnhead,et al.  On the Asymptotic Efficiency of ABC Estimators , 2015 .

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

[27]  Jean-Jacques Forneron,et al.  A Likelihood-Free Reverse Sampler of the Posterior Distribution , 2015, 1506.04017.

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

[29]  Michele Haynes.  Flexible distributions and statistical models in ranking and selection procedures with applications , 1998 .

[30]  Van Der Vaart,et al.  The Bernstein-Von-Mises theorem under misspecification , 2012 .

[31]  S. Wood.  Statistical inference for noisy nonlinear ecological dynamic systems , 2010, Nature.

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

[33]  Mike West,et al.  Sequential Monte Carlo with Adaptive Weights for Approximate Bayesian Computation , 2015, 1503.07791.

[34]  C C Drovandi,et al.  Estimation of Parameters for Macroparasite Population Evolution Using Approximate Bayesian Computation , 2011, Biometrics.

[35]  David T. Frazier,et al.  Asymptotic properties of approximate Bayesian computation , 2016, Biometrika.

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

[37]  V. Chernozhukov,et al.  An MCMC Approach to Classical Estimation , 2002, 2301.07782.

[38]  N. Reid,et al.  AN OVERVIEW OF COMPOSITE LIKELIHOOD METHODS , 2011 .