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

TL;DR

This paper analyzes the frequentist coverage of Bayesian credible sets in nonlinear inverse PDE problems with Gaussian priors, ensuring valid inference under certain conditions.

Contribution

It establishes conditions under which Bayesian credible intervals have conservative frequentist coverage in nonlinear inverse problems, even without efficiency or BvM theorem.

Findings

Bayes credible intervals can be conservative under smoothness and compatibility conditions.

Results apply to PDE inverse problems like Darcy flow, with near-$1/ oot{N}$ contraction rates.

Linear functionals not estimable efficiently still achieve conservative coverage.

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.