A1132
Title: Valid uncertainty quantification for linear functionals in semi-parametric regression models
Authors: Gustav Romer - University of Cambridge (Denmark) [presenting]
Abstract: Inspired by Darcy's problem, the frequentist validity of Bayesian uncertainty quantification is addressed for irregular linear functionals in semi-parametric models. For a finite collection of linear functionals, a renormalized Bernstein-von-Mises theorem is proven to allow for posterior credible sets that are asymptotic confidence sets. This is demonstrated for a credible ellipsoid centered at the posterior mean and shaped by the posterior variance. A bound is provided on its diameter, and a Wald-type ellipsoid is introduced as an alternative. For a single linear function, results are obtained for the symmetric credible interval around the posterior mean. A general inverse problem is then analyzed, represented by a Gaussian regression model, where the regression functions are parameterized by a non-linear forward map. A high-dimensional Gaussian prior is employed. Assuming the invertibility of the information matrix in high-dimensional approximation models, a renormalized Bernstein-von-Mises theorem is established for a finite collection of linear functionals. The conditions for the forward map induced by the partial differential equation, Darcy's problem, where irregular linear functionals naturally arise, are explicitly checked.
