Posterior Contraction in Bayesian Inverse Problems Under Gaussian Priors

Abstract

We study Bayesian inference in statistical linear inverse problems with Gaussian noise and priors in a separable Hilbert space setting. We focus our interest on the posterior contraction rate in the small noise limit, under the frequentist assumption that there exists a fixed data-generating value of the unknown. In this Gaussian-conjugate setting, it is convenient to work with the concept of squared posterior contraction (SPC), which is known to upper bound the posterior contraction rate. We use abstract tools from regularization theory, which enable a unified approach to bounding SPC. We review and re-derive several existing results, and establish minimax contraction rates in cases which have not been considered until now. Existing results suffer from a certain saturation phenomenon, when the data-generating element is too smooth compared to the smoothness inherent in the prior. We show how to overcome this saturation in an empirical Bayesian framework by using a non-centered data-dependent prior.