Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors

By F. Monard, Richard Nickl, Gabriel P. Paternain. Annals of Statistics, 2020

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Bayesian inference and uncertainty quantification in a general class of non-linear inverse regression models is considered. Analytic conditions on the regression model $\{\mathscr G(\theta): \theta \in \Theta\}$ 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 shown to be valid and optimal from a frequentist point of view. The theory is demonstrated to cover two prototypical applications with PDEs that arise in non-linear tomography problems: the first concerns an elliptic inverse problem for the Schr\"odinger equation, and the second the inversion of non-Abelian $X$-ray transforms. New PDE techniques are developed to show that the relevant Fisher information operators are invertible between suitable function spaces.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/05082af09a90f113070b1d4ad2f5e53df23fed47