Theory and Practice of Bayesian Methods in Inverse Problems and Related Nonparametric Models

Abstract

This dissertation collects results on novel applications of Bayesian inference and computation to areas using high-to-infinite-dimensional mathematical models. The two areas covered here are inverse problems based on mathematical modeling through partial differential equations and probabilistic machine learning through deep neural networks. Chapter \ref{chap:freq-inverse} presents fundamental results on the frequentist coverage of Bayes posteriors in nonlinear inverse problems based on PDE models. Chapters \ref{chap:uq-misspec} presents theoretical and methodological results on using generalized Bayes posteriors in these inverse problems under model misspecification. Chapter \ref{chap:bayes-rf} presents results on the generalizability of posterior inference on new examples in a 2-layer neural network setting. We present new theoretical results on the frequentist coverage of Bayes posterior high probability sets corresponding to reasonable Gaussian process priors in each problem and present a methodological and computational workflow for Bayesian inverse problems when the likelihood model is potentially misspecified.