The rate of convergence for approximate Bayesian computation

Abstract

Approximate Bayesian Computation (ABC) is a popular computational method for likelihood-free Bayesian inference. The term "likelihoodfree" 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 tolerance 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 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 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. keywords: Approximate Bayesian Computation, likelihood-free inference, Monte Carlo methods, convergence of estimators, rate of convergence MSC2010 classes: 62F12 (Asymptotic properties of estimators), 62F15 (Bayesian inference), 65C05 (Monte Carlo methods)

Bayesian estimation of agent-based models

“…In particular, it can be shown that the bias of the ABC estimate is asymptotically proportional to h 2 , as h → 0. On the other hand, the computational cost for generating n accepted ABC samples is proportional to nh −q , where q is the dimension of the observations (Barber et al, 2015). The same authors show that the mean squared error (MSE) of the estimates for constant expected computational costs is given by…”

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“…The discrepancy threshold determines the level of approximation, however, under the assumption of model and observation error, Equation (31) can be treated as exact [151]. [9,40]. Using data for the monomolecular chain model, we can demonstrate this convergence, as shown in Figure 4 (see also Appendix D).…”

“…Through application of the ABC with acceptance threshold, , the equivalent marginals are p(k 1 | ρ(Y obs , S obs ) ≤ ) = Reducing further than 12.5 is prohibitive, even for the mono-molecular chain model. Both Barber et al [9] and Fearnhead and Prangle [40] provide an asymptotic result for the computation time, C, as a function of , that is, C = O( −d ), where d is the dimensionality of the data used in the ABC inference. For the synthetic data we have from Table C.1, we have d = n t N .…”

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“…A formal treatment of the issue is given by Blum (2010), Barber et al (2015) and Biau et al (2015) amongst others.…”