Electronic Journal of Statistics Vol. 9 (2015) 80–105 ISSN: 1935-7524 DOI: 10.1214/15-EJS988 The rate of convergence for approximate Bayesian computation Stuart Barber, Jochen Voss and Mark Webster School of Mathematics University of Leeds Leeds LS2 9JT, UK e-mail: s.barber@leeds.ac.uk ; j.voss@leeds.ac.uk ; mm08mgw@leeds.ac.uk Abstract: Approximate Bayesian Computation (ABC) is a popular com- putational method for likelihood-free Bayesian inference. The term “likeli- hood-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 θ simulated from a prior distribution. Parameters which generate simulated data within some toler- ance δ of the observed data x * are regarded as plausible, and a collection of such θ is used to estimate the posterior distribution θ | X = x * . Suitable choice of δ is vital for ABC methods to return good approximations to θ in reasonable computational time. While ABC methods are widely used in practice, particularly in popula- tion genetics, rigorous study of the mathematical properties of ABC estima- tors 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 δ 2 as δ ↓ 0. At the same time, the computational cost for generating one ABC sample increases like δ -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 δ. MSC 2010 subject classifications: Primary 62F12, 65C05; secondary 62F15. Keywords and phrases: Approximate Bayesian computation, likelihood- free inference, Monte Carlo methods, convergence of estimators, rate of convergence. Received August 2014. 1. Introduction Approximate Bayesian Computation (ABC) is a popular method for likelihood- free Bayesian inference. ABC methods were originally introduced in popula- tion genetics, but are now widely used in applications as diverse as epidemiol- ogy (Tanaka et al., 2006; Blum and Tran, 2010; Walker et al., 2010), materials science (Bortot et al., 2007), parasitology (Drovandi and Pettitt, 2011), genetic evolution (Thornton and Andolfatto, 2006; Fagundes et al., 2007; Ratmann et al., 2009; Wegmann and Excoffier, 2010; Beaumont, 2010; Wilkinson et al., 2011), 80