  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.
