Frequentist Coverage of Bayes Posteriors in Nonlinear Inverse Problems with Gaussian Priors

Abstract

We study asymptotic frequentist coverage and approximately Gaussian properties of Bayes posterior credible sets in nonlinear inverse problems when a Gaussian prior is placed on the parameter of the PDE. The aim is to ensure valid frequentist coverage of Bayes credible intervals when estimating continuous linear functionals of the parameter. Our results show that Bayes credible intervals have conservative coverage under certain smoothness assumptions on the parameter and a compatibility condition between the likelihood and the prior, regardless of whether an efficient limit exists or Bernstein von-Mises (BvM) theorem holds. In the latter case, our results yield a corollary with more relaxed sufficient conditions than previous works. The theory is illustrated with a PDE that arises in predicting the transport of radioactive waste from underground repositories and optimizing oil recovery from subsurface fields: an elliptic inverse problem for Darcy flow. In this case, a near-$1/\sqrt{N}$ contraction rate and conservative coverage results are obtained for linear functionals that were shown not to be estimable efficiently.