Deep Generative Quantile Bayes

TL;DR

This paper introduces a deep generative quantile learning method for Bayesian posterior sampling that leverages multivariate quantiles and neural networks, enabling credible set sampling without likelihood-based methods.

Contribution

It presents a novel posterior sampling approach using deep generative quantile learning with Monge-Kantorovich depth, including neural network-based summary statistics and theoretical guarantees.

Findings

Supports shrinkage of posterior approximation as data increases

Enables sampling from credible sets without likelihood functions

Demonstrates effectiveness on examples where MCMC fails

Abstract

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 vector 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 the 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.