  This paper proposes a new methodology for performing Bayesian inference in
imaging inverse problems where the prior knowledge is available in the form of
training data. Following the manifold hypothesis and adopting a generative
modelling approach, we construct a data-driven prior that is supported on a
sub-manifold of the ambient space, which we can learn from the training data by
using a variational autoencoder or a generative adversarial network. We
establish the existence and well-posedness of the associated posterior
distribution and posterior moments under easily verifiable conditions,
providing a rigorous underpinning for Bayesian estimators and uncertainty
quantification analyses. Bayesian computation is performed by using a parallel
tempered version of the preconditioned Crank-Nicolson algorithm on the
manifold, which is shown to be ergodic and robust to the non-convex nature of
these data-driven models. In addition to point estimators and uncertainty
quantification analyses, we derive a model misspecification test to
automatically detect situations where the data-driven prior is unreliable, and
explain how to identify the dimension of the latent space directly from the
training data. The proposed approach is illustrated with a range of experiments
with the MNIST dataset, where it outperforms alternative image reconstruction
approaches from the state of the art. A model accuracy analysis suggests that
the Bayesian probabilities reported by the data-driven models are also
remarkably accurate under a frequentist definition of probability.
