Generalized Bayesian Likelihood-Free Inference Using Scoring Rules Estimators

We propose a framework for Bayesian Likelihood-Free Inference (LFI) based on Generalized Bayesian Inference using scoring rules (SR). SR are used to evaluate probabilistic models given an observation; a proper SR is minimised in expectation when the model corresponds to the true data generating process for the observation. Using a strictly proper SR, for which the above minimum is unique, ensures posterior consistency of our method. As the likelihood function is intractable for LFI, we employ consistent estimators of SR using model simulations in a pseudo-marginal MCMC; we show the target of such chain converges to the exact SR posterior with increasing number of simulations. Furthermore, we note popular LFI techniques like Bayesian Synthetic Likelihood (BSL) and semiparametric BSL can be seen as special cases of our framework using only proper (but not strictly so) SR. We provide empirical results validating our consistency result and show how related approaches do not enjoy this property. Practically, we use the Energy and Kernel Scores, but our general framework sets the stage for extensions with other scoring rules.