Generalized Bayes Approach to Inverse Problems with Model Misspecification

TL;DR

This paper introduces a flexible Bayesian-inspired framework for inverse problems that handles model misspecification by directly optimizing a variational objective and evaluating loss functions through predictive performance, demonstrated on ultrasound vibrometry data.

Contribution

It develops a Gibbs posterior-based approach for inverse problems that does not rely on explicit likelihood models and includes a novel model comparison and calibration method.

Findings

Effective in a simulated arterial vessel characterization example

Provides a new way to evaluate loss functions via cross-validation

Theoretical properties of Gibbs posteriors are established

Abstract

We propose a general framework for obtaining probabilistic solutions to PDE-based inverse problems. Bayesian methods are attractive for uncertainty quantification but assume knowledge of the likelihood model or data generation process. This assumption is difficult to justify in many inverse problems, where the specification of the data generation process is not obvious. We adopt a Gibbs posterior framework that directly posits a regularized variational problem on the space of probability distributions of the parameter. We propose a novel model comparison framework that evaluates the optimality of a given loss based on its "predictive performance". We provide cross-validation procedures to calibrate the regularization parameter of the variational objective and compare multiple loss functions. Some novel theoretical properties of Gibbs posteriors are also presented. We illustrate the utility of our framework via a simulated example, motivated by dispersion-based wave models used to characterize arterial vessels in ultrasound vibrometry.