Bernstein–von Mises theorems for statistical inverse problems I: Schrödinger equation
Richard Nickl
Abstract
We consider the inverse problem of determining the potential f > 0 in the partial differential equation \frac{\Delta}{2} u - fu =0 \: \mathrm{on}\: \mathcal O, \:\: u = g \: \mathrm{on} \: \partial \mathcal O, where \mathcal O is a bounded C^\infty -domain in \mathbb R^d and g > 0 is a given function prescribing boundary values. The data consist of the solution u corrupted by additive Gaussian noise. A nonparametric Bayesian prior for the function f is devised and a Bernstein–von Mises theorem is proved which entails that the posterior distribution given the observations is approximated in a suitable function space by an infinite-dimensional Gaussian measure that has a ‘minimal’ covariance structure in an information-theoretic sense. As a consequence the posterior distribution performs valid and optimal frequentist statistical inference on various aspects of f in the small noise limit.