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Approximate Bayesian Computation (ABC) methods rely on asymptotic arguments,implying that parameter inference can be systematically biased even whensufficient statistics are available. We propose to construct the ABCaccept/reject step from decision theoretic arguments on a suitable auxiliaryspace. This framework, referred to as ABC*, fully specifies which teststatistics to use, how to combine them, how to set the tolerances and how longto simulate in order to obtain accuracy properties on the auxiliary space. Akinto maximum-likelihood indirect inference, regularity conditions establish whenthe ABC* approximation to the posterior density is accurate on the originalparameter space in terms of the Kullback-Leibler divergence and the maximum aposteriori point estimate. Fundamentally, escaping asymptotic argumentsrequires knowledge of the distribution of test statistics, which we obtainthrough modelling the distribution of summary values, data points on a summarylevel. Synthetic examples and an application to time series data of influenza A(H3N2) infections in the Netherlands illustrate ABC* in action.

Statistical modelling of summary values leads to accurate Approximate Bayesian Computations