Approximate Bayesian Computation: a nonparametric perspective

Approximate Bayesian Computation is a family of likelihood-free inference techniques that are tailored to models defined in terms of a stochastic generating mechanism. In a nutshell, Approximate Bayesian Computation proceeds by computing summary statistics from the data and giving more weight to the values of the parameters for which the simulated summary statistics resemble the observed ones. In this paper, we present Approximate Bayesian Computation as a technique of inference that relies on stochastic simulations and non-parametric statistics. We derive the asymptotic bias and variance of the standard estimators of the posterior distribution which are based on rejection sampling and linear adjustment. Additionally, we introduce an original estimator of the posterior distribution based on quadratic adjustment and we show that its bias contains a smaller number of terms than the estimator with linear adjustment. Although we find that the estimators with adjustment are not universally superior to the estimator based on rejection sampling, we find that they can achieve better performance when there is a nearly homoscedastic relationship between the summary statistics and the parameter of interest. Last, we present model selection in Approximate Bayesian Computation and provide asymptotic properties of two estimators of the model probabilities. As for parameter estimation, the asymptotic results raise the importance of the curse of dimensionality in Approximate Bayesian Computation. Performing numerical simulations, in a simple normal model, confirms that the quality of the estimators deteriorates as the number of summary statistics increases.