Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors
François Monard, Richard Nickl, Gabriel P. Paternain
The Annals of Statistics · 2021 · Volume 49 · Issue 6
Abstract
Bayesian inference and uncertainty quantification in a general class of nonlinear inverse regression models is considered. Analytic conditions on the regression model {G(θ):θ∈Θ} and on Gaussian process priors for θ are provided such that semiparametrically efficient inference is possible for a large class of linear functionals of θ. A general 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 nonlinear tomography problems: an elliptic inverse problem for a Schrödinger 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.
Keywords
62F15, 65N21, Bernstein–von Mises theorems, credible sets, Schrödinger equation, X-ray transforms
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