Approximate Bayesian computation using asymptotically normal point estimates

Approximate Bayesian computation (ABC) provides inference of the posterior distribution, even for models with intractable likelihoods, by replacing the exact (intractable) model likelihood by a tractable approximate likelihood. Meanwhile, historically, the development of point-estimation methods usually precedes the development of posterior estimation methods. We propose and study new ABC methods based on asymptotically normal and consistent point-estimators of the model parameters. Specifically, for the classical ABC method, we propose and study two alternative bootstrap methods for estimating the tolerance tuning parameter, based on resampling from the asymptotic normal distribution of the given point-estimator. This tolerance estimator can be quickly computed even for any model for which it is computationally costly to sample directly from its exact likelihood, provided that its summary statistic is specified as consistent point-estimator of the model parameters with estimated asymptotic normal distribution that can typically be easily sampled from. Furthermore, this paper introduces and studies a new ABC method based on approximating the exact intractable likelihood by the asymptotic normal density of the point-estimator, motivated by the Bernstein-Von Mises theorem. Unlike the classical ABC method, this new approach does not require tuning parameters, aside from the summary statistic (the parameter point estimate). Each of the new ABC methods is illustrated and compared through a simulation study of tractable models and intractable likelihood models, and through the Bayesian intractable likelihood analysis of a real 23,000-node network dataset involving stochastic search model selection.