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Bernstein-von Mises theorems and uncertainty quantification for linear inverse problems

Abstract

We consider the statistical inverse problem of approximating an unknown function ff from a linear measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the posterior-based reconstruction of ff corresponds to a Tikhonov regulariser fˉ\bar f with a Cameron-Martin space norm penalty. We prove a semiparametric Bernstein-von Mises theorem for a large collection of linear functionals of ff, implying that semiparametric posterior estimation and uncertainty quantification are valid and optimal from a frequentist point of view. The result is illustrated and further developed for some examples both in mildly and severely ill-posed cases. For the problem of recovering the source function in elliptic partial differential equations, we also obtain a nonparametric version of the theorem that entails the convergence of the posterior distribution to a fixed infinite-dimensional Gaussian probability measure with minimal covariance in suitable function spaces. As a consequence, we show that the distribution of the Tikhonov regulariser fˉ\bar f is asymptotically normal and attains the information lower bound, and that credible sets centred at fˉ\bar{f} have correct frequentist coverage and optimal diameter.

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