Statistical guarantees for Bayesian uncertainty quantification in non-linear inverse problems with Gaussian process priors
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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 {𝒢(θ): θ∈Θ} and on Gaussian process priors for θ are provided such that semi-parametrically efficient inference is possible for a large class of linear functionals of θ. 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 Schrödinger 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.
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