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Bayesian Likelihood-Free Inference methods yield posterior approximations forsimulator models with intractable likelihood. Recently, many works trainedneural networks to approximate either the intractable likelihood or theposterior directly. Most proposals use normalizing flows, namely neuralnetworks parametrizing invertible maps used to transform samples from anunderlying base measure; the probability density of the transformed samples isthen accessible and the normalizing flow can be trained via maximum likelihoodon simulated parameter-observation pairs. A recent work [Ramesh et al., 2022]approximated instead the posterior with generative networks, which drop theinvertibility requirement and are thus a more flexible class of distributionsscaling to high-dimensional and structured data. However, generative networksonly allow sampling from the parametrized distribution; for this reason, Rameshet al. [2022] follows the common solution of adversarial training, where thegenerative network plays a min-max game against a "critic" network. Thisprocedure is unstable and can lead to a learned distribution underestimatingthe uncertainty - in extreme cases collapsing to a single point. Here, wepropose to approximate the posterior with generative networks trained byScoring Rule minimization, an overlooked adversarial-free method enablingsmooth training and better uncertainty quantification. In simulation studies,the Scoring Rule approach yields better performances with shorter training timewith respect to the adversarial framework.

Likelihood-Free Inference with Generative Neural Networks via Scoring Rule Minimization