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Approximate Bayesian computation (ABC) is a widely used inference method inBayesian statistics to bypass the point-wise computation of the likelihood. Inthis paper we develop theoretical bounds for the distance between thestatistics used in ABC. We show that some versions of ABC are inherently robustto misspecification. The bounds are given in the form of oracle inequalitiesfor a finite sample size. The dependence on the dimension of the parameterspace and the number of statistics is made explicit. The results are shown tobe amenable to oracle inequalities in parameter space. We apply our theoreticalresults to given prior distributions and data generating processes, including anon-parametric regression model. In a second part of the paper, we propose asequential Monte Carlo (SMC) to sample from the pseudo-posterior, improvingupon the state of the art samplers.

Probably approximate Bayesian computation: nonasymptotic convergence of ABC under misspecification