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This paper proposes a new methodology for performing Bayesian inference inimaging inverse problems where the prior knowledge is available in the form oftraining data. Following the manifold hypothesis and adopting a generativemodelling approach, we construct a data-driven prior that is supported on asub-manifold of the ambient space, which we can learn from the training data byusing a variational autoencoder or a generative adversarial network. Weestablish the existence and well-posedness of the associated posteriordistribution and posterior moments under easily verifiable conditions,providing a rigorous underpinning for Bayesian estimators and uncertaintyquantification analyses. Bayesian computation is performed by using a paralleltempered version of the preconditioned Crank-Nicolson algorithm on themanifold, which is shown to be ergodic and robust to the non-convex nature ofthese data-driven models. In addition to point estimators and uncertaintyquantification analyses, we derive a model misspecification test toautomatically detect situations where the data-driven prior is unreliable, andexplain how to identify the dimension of the latent space directly from thetraining data. The proposed approach is illustrated with a range of experimentswith the MNIST dataset, where it outperforms alternative image reconstructionapproaches from the state of the art. A model accuracy analysis suggests thatthe Bayesian probabilities reported by the data-driven models are alsoremarkably accurate under a frequentist definition of probability.

Bayesian Imaging With Data-Driven Priors Encoded by Neural Networks: Theory, Methods, and Algorithms