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Statistical guarantees for Bayesian uncertainty quantification in inverse problems

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UNQW04 - UQ for inverse problems in complex systems

We discuss recent results in mathematical statistics that provide objective statistical guarantees for Bayesian algorithms in (possibly non-linear) noisy inverse problems. We focus in particular on the justification of Bayesian credible sets as proper frequentist confidence sets in the small noise limit via so-called `Bernstein – von Mises theorems', which provide Gaussian approximations to the posterior distribution, and introduce notions of such theorems in the infinite-dimensional settings relevant for inverse problems. We discuss in detail such a Bernstein-von Mises result for Bayesian inference on the unknown potential in the Schroedinger equation from an observation of the solution of that PDE corrupted by additive Gaussian white noise. See https://arxiv.org/abs/1707.01764 and also https://arxiv.org/abs/1708.06332

This talk is part of the Isaac Newton Institute Seminar Series series.

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