Electronic Journal of Statistics

The rate of convergence for approximate Bayesian computation

Stuart Barber, Jochen Voss, and Mark Webster

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Approximate Bayesian Computation (ABC) is a popular computational method for likelihood-free Bayesian inference. The term “likelihood-free” refers to problems where the likelihood is intractable to compute or estimate directly, but where it is possible to generate simulated data $X$ relatively easily given a candidate set of parameters $\theta$ simulated from a prior distribution. Parameters which generate simulated data within some tolerance $\delta$ of the observed data $x^{*}$ are regarded as plausible, and a collection of such $\theta$ is used to estimate the posterior distribution $\theta |X=x^{*}$. Suitable choice of $\delta$ is vital for ABC methods to return good approximations to $\theta$ in reasonable computational time.

While ABC methods are widely used in practice, particularly in population genetics, rigorous study of the mathematical properties of ABC estimators lags behind practical developments of the method. We prove that ABC estimates converge to the exact solution under very weak assumptions and, under slightly stronger assumptions, quantify the rate of this convergence. In particular, we show that the bias of the ABC estimate is asymptotically proportional to $\delta^{2}$ as $\delta\downarrow 0$. At the same time, the computational cost for generating one ABC sample increases like $\delta^{-q}$ where $q$ is the dimension of the observations. Rates of convergence are obtained by optimally balancing the mean squared error against the computational cost. Our results can be used to guide the choice of the tolerance parameter $\delta$.

Article information

Electron. J. Statist., Volume 9, Number 1 (2015), 80-105.

Received: August 2014
First available in Project Euclid: 6 February 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 65C05: Monte Carlo methods
Secondary: 62F15: Bayesian inference

Approximate Bayesian computation likelihood-free inference Monte Carlo methods convergence of estimators rate of convergence


Barber, Stuart; Voss, Jochen; Webster, Mark. The rate of convergence for approximate Bayesian computation. Electron. J. Statist. 9 (2015), no. 1, 80--105. doi:10.1214/15-EJS988. https://projecteuclid.org/euclid.ejs/1423229751

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  • Barber, S., Voss, J., and Webster, M. (2015). Supplement to “The rate of convergence for approximate Bayesian computation”. DOI:10.1214/, 15-EJS988SUPP.
  • Beaumont, M. A., Approximate Bayesian computation in evolution and ecology., Annual Review of Ecology, Evolution, and Systematics, 41:379–406, December 2010.
  • Biau, G., Cérou, F., and Guyader, A., New insights into approximate Bayesian computation., Annales de l’Institut Henri Poincaré, 2013. In press.
  • Blum, M. G. B., Approximate Bayesian computation: A nonparametric perspective., Journal of the American Statistical Association, 105(491) :1178–1187, September 2010.
  • Blum, M. G. B. and François, O., Non-linear regression models for Approximate Bayesian Computation., Statistics and Computing, 20:63–73, 2010.
  • Blum, M. G. B. and Tran, V.-C., HIV with contact tracing: a case study in approximate Bayesian computation., Biostatistics, 11(4):644–660, 2010.
  • Bortot, P., Coles, S. G., and Sisson, S. A., Inference for stereological extremes., Journal of the American Statistical Association, 102(477):84–92, 2007.
  • Dembo, A. and Zeitouni, O., Large Deviations Techniques and Applications, volume 38 of Applications of Mathematics. Springer, second edition, 1998.
  • Drovandi, C. C. and Pettitt, A. N., Estimation of parameters for macroparasite population evolution using approximate Bayesian computation., Biometrics, 67(1):225–233, 2011.
  • Fagundes, N. J. R., Ray, N., Beaumont, M., Neuenschwander, S., Salzano, F. M., Bonatto, S. L., and Excoffier, L., Statistical evaluation of alternative models of human evolution., Proceedings of the National Academy of Sciences, 104(45) :17614–17619, 2007.
  • Fearnhead, P. and Prangle, D., Constructing summary statistics for approximate Bayesian computation: Semi-automatic approximate Bayesian computation., Journal of the Royal Statistical Society: Series B, 74(3):419–474, 2012.
  • Guillemaud, T., Beaumont, M. A., Ciosi, M., Cornuet, J.-M., and Estoup, A., Inferring introduction routes of invasive species using approximate Bayesian computation on microsatellite data., Heredity, 104(1):88–99, 2010.
  • Lopes, J. S. and Boessenkool, S., The use of approximate Bayesian computation in conservation genetics and its application in a case study on yellow-eyed penguins., Conservation Genetics, 11(2):421–433, 2010.
  • Marin, J.-M., Pudlo, P., Robert, C. P., and Ryder, R. J., Approximate Bayesian computational methods., Statistics and Computing, 22(6) :1167–1180, 2012.
  • Marjoram, P., Molitor, J., Plagnol, V., and Tavaré, S., Markov chain Monte Carlo without likelihoods., Proceedings of the National Academy of Sciences, 100(26) :15324–15328, 2003.
  • R Core Team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2013.
  • Ratmann, O., Andrieu, C., Wiuf, C., and Richardson, S., Model criticism based on likelihood-free inference, with an application to protein network evolution., Proceedings of the National Academy of Sciences, 106(26) :10576–10581, 2009.
  • Rudin, W., Real and Complex Analysis. McGraw-Hill, third edition, 1987.
  • Silk, D., Filippi, S., and M. Stumpf, P. H., Optimizing threshold-schedules for sequential approximate Bayesian computation: Applications to molecular systems., Statistical Applications in Genetics and Molecular Biology, 12(5):603–618, September 2013.
  • Sisson, S., Fan, Y., and Tanaka, M. M., Sequential Monte Carlo without likelihoods., Proceedings of the National Academy of Sciences, 104(6) :1760–1765, 2007.
  • Sottoriva, A. and Tavaré, S., Integrating approximate Bayesian computation with complex agent-based models for cancer research. In Y. Lechevallier and G. Saporta, editors, Proceedings of COMPSTAT ’2010, pages 57–66. Springer, 2010.
  • Tanaka, M. M., Francis, A. R., Luciani, F., and Sisson, S. A., Using approximate Bayesian computation to estimate tuberculosis transmission parameters from genotype data., Genetics, 173(3) :1511–1520, 2006.
  • Tavaré, S., Balding, D. J., Griffiths, R. C., and Donnelly, P., Inferring coalescence times from DNA sequence data., Genetics, 145(2):505–518, 1997.
  • Thornton, K. and Andolfatto, P., Approximate Bayesian inference reveals evidence for a recent, severe bottleneck in a Netherlands population of, Drosophila melanogaster. Genetics, 172(3) :1607–1619, 2006.
  • Voss, J., An Introduction to Statistical Computing: A Simulation-Based Approach. Wiley Series in Computational Statistics. Wiley, 2014. ISBN 978-1118357729.
  • Walker, D. M., Allingham, D., Lee, H. W. J., and Small, M., Parameter inference in small world network disease models with approximate Bayesian computational methods., Physica A, 389(3):540–548, 2010.
  • Wegmann, D. and Excoffier, L., Bayesian inference of the demographic history of chimpanzees., Molecular Biology and Evolution, 27(6) :1425–1435, 2010.
  • Wilkinson, R. D., Approximate Bayesian computation (ABC) gives exact results under the assumption of model error., Statistical Applications in Genetics and Molecular Biology, 12(2):129–141, 2013.
  • Wilkinson, R. D., Steiper, M. E., Soligo, C., Martin, R. D., Yang, Z., and Tavaré, S., Dating primate divergences through an integrated analysis of palaeontological and molecular data., Systematic Biology, 60(1):16–31, 2011.

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