On the Frequentist Coverage of Bayes Posteriors in Nonlinear Inverse Problems

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 and/or Bernstein von-Mises theorem holds. In the latter case, our results yield a corollary with more relaxed sufficient conditions than previous works. We illustrate practical utility of the results through the example of estimating the conductivity coefficient of a second order elliptic PDE, where a near-$N^{-1/2}$ contraction rate and conservative coverage results are obtained for linear functionals that were shown not to be estimable efficiently.