Statistical guarantees for Bayesian uncertainty quantification in non-linear inverse problems with Gaussian process priors

Bayesian inference and uncertainty quantification in a general class of non-linear inverse regression models is considered. Analytic conditions on the regression model $\{\mathcal{G}(\theta ):\theta \in \mathrm{\Theta }\}$ and on Gaussian process priors for $\theta$ are provided such that semi-parametrically efficient inference is possible for a large class of linear functionals of $\theta$. A general semi-parametric Bernstein-von Mises theorem is proved that shows that the (non-Gaussian) posterior distributions are approximated by certain Gaussian measures centred at the posterior mean. As a consequence posterior-based credible sets are valid and optimal from a frequentist point of view. The theory is illustrated with two applications with PDEs that arise in non-linear tomography problems: an elliptic inverse problem for a Schr"odinger equation, and inversion of non-Abelian X-ray transforms. New analytical techniques are deployed to show that the relevant Fisher information operators are invertible between suitable function spaces