  In the absence of explicit or tractable likelihoods, Bayesians often resort
to approximate Bayesian computation (ABC) for inference. Our work bridges ABC
with deep neural implicit samplers based on generative adversarial networks
(GANs) and adversarial variational Bayes. Both ABC and GANs compare aspects of
observed and fake data to simulate from posteriors and likelihoods,
respectively. We develop a Bayesian GAN (B-GAN) sampler that directly targets
the posterior by solving an adversarial optimization problem. B-GAN is driven
by a deterministic mapping learned on the ABC reference by conditional GANs.
Once the mapping has been trained, iid posterior samples are obtained by
filtering noise at a negligible additional cost. We propose two post-processing
local refinements using (1) data-driven proposals with importance reweighting,
and (2) variational Bayes. We support our findings with frequentist-Bayesian
results, showing that the typical total variation distance between the true and
approximate posteriors converges to zero for certain neural network generators
and discriminators. Our findings on simulated data show highly competitive
performance relative to some of the most recent likelihood-free posterior
simulators.
