Deep Generative Quantile Bayes

We develop a multivariate posterior sampling procedure through deep generative quantile learning. Simulation proceeds implicitly through a push-forward mapping that can transform i.i.d. random vectors samples from the posterior. We utilize Monge-Kantorovich depth in multivariate quantiles to directly sample from Bayesian credible sets, a unique feature not offered by typical posterior sampling methods. To enhance training of the quantile mapping, we design a neural network that automatically performs summary statistic extraction. This additional neural network structure has performance benefits including support shrinkage (i.e. contraction of our posterior approximation) as the observation sample size increases. We demonstrate the usefulness of our approach on several examples where the absence of likelihood renders classical MCMC infeasible. Finally, we provide the following frequentist theoretical justifications for our quantile learning framework: consistency of the estimated vector quantile, of the recovered posterior distribution, and of the corresponding Bayesian credible sets.