Generalized Bayes approach to inverse problems with model misspecification.

Area of Science:

• Computational Mathematics
• Applied Mathematics
• Scientific Computing

Background:

• Bayesian methods are standard for uncertainty quantification in inverse problems.
• They require accurate likelihood models, which are often unavailable or difficult to specify.
• This limits their application in real-world scenarios where data generation processes are complex.

Purpose of the Study:

• To develop a general framework for probabilistic solutions to PDE-based inverse problems.
• To address the challenge of unknown likelihood models in Bayesian inference.
• To introduce a novel model comparison framework based on predictive performance.

Main Methods:

• Utilizing a Gibbs posterior framework with a regularized variational problem on the space of probability distributions.
• Developing a model comparison framework to evaluate loss function optimality via predictive performance.
• Implementing cross-validation for regularization parameter calibration and loss function comparison.

Main Results:

• Demonstrated a novel probabilistic framework for PDE-based inverse problems.
• Presented theoretical properties of Gibbs posteriors.
• Validated the framework using a simulated example from ultrasound vibrometry for arterial vessel characterization.

Conclusions:

• The proposed framework offers a robust alternative for uncertainty quantification when likelihood models are unknown.
• The model comparison approach facilitates optimal loss function selection.
• The method shows promise for applications in medical imaging and other fields.