We will consider a family of potentially nonlinear inverse problems subject to Gaussian additive white noise. We will assume truncated Gaussian priors and our interest will be in studying the asymptotic performance of the Bayesian posterior in the small noise limit. In particular, we will develop a theory for obtaining posterior contraction rates. The theory is based on the techniques of Knapik and Salomond 2018, which show how to derive posterior contraction rates for inverse problems, using rates of contraction for direct problems and the notion of the modulus of continuity. We will work under the assumption that the forward operator can be associated to a linear operator in a certain sense. We will present techniques from regularization theory, which allow both to bound the modulus of continuity, as well as to derive optimal rates of contraction for the direct problem by appropriately tuning the prior-truncation level. Finally, we will combine to obtain optimal rates of contraction for a range of inverse problems.
This is joint work with Peter Mathé (Weierstrass Institute, Berlin)