  Bayesian inference is used extensively to quantify the uncertainty in an
inferred field given the measurement of a related field when the two are linked
by a mathematical model. Despite its many applications, Bayesian inference
faces challenges when inferring fields that have discrete representations of
large dimension, and/or have prior distributions that are difficult to
characterize mathematically. In this work we demonstrate how the approximate
distribution learned by a deep generative adversarial network (GAN) may be used
as a prior in a Bayesian update to address both these challenges. We
demonstrate the efficacy of this approach on two distinct, and remarkably
broad, classes of problems. The first class leads to supervised learning
algorithms for image classification with superior out of distribution detection
and accuracy, and for image inpainting with built-in variance estimation. The
second class leads to unsupervised learning algorithms for image denoising and
for solving physics-driven inverse problems.
