Generalized Bayes Approach to Inverse Problems with Model Misspecification

Abstract

I discuss a general framework for obtaining probabilistic solutions to PDE-based inverse problems when potentially the PDE is inaccurate or the noise-generating mechanism is unknown. In a generalized Bayesian formulation, the Bayesian update problem is reformulated and generalized into a regularized variational problem on the space of probability distributions of the parameter. A novel generalization of a Bayesian model comparison procedure is given for evaluating the optimality of a given loss based on its “predictive performance.” A tailored sequential Monte Carlo-based approach is used to simultaneously calibrate the regularization parameter and obtain samples from the underlying posterior. Some theoretical properties of Gibbs posteriors are also presented.

Advisor(s)

Sayan Mukherjee