Probabilistic Approach to Parameteric Inverse Problems Using Gibbs Posteriors

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, in which the random map from the parameters to the data is complex and nonlinear. We adopt a Gibbs posterior framework that directly posits a regularized variational problem on the space of probability distributions of the parameter. The minimizing solution generalizes the Bayes posterior and is robust to misspecification or lack of knowledge of the generative model. We provide cross-validation procedures to set the regularization hyperparameter in our inference framework. A practical and computational advantage of our framework is that its implementation draws on existing tools in Bayesian computational statistics. We illustrate the utility of our framework via a simulated example, motivated by dispersion-based wave models used to characterize artertial vessels in ultrasound vibrometry.