Formulation and properties of a divergence used to compare probability measures without absolute continuity and its application to uncertainty quantification

TL;DR

This paper introduces a new divergence measure for comparing probability distributions without requiring absolute continuity, with applications in uncertainty quantification and a connection to optimal transport and relative entropy.

Contribution

It develops a generalized divergence that extends relative entropy, providing new theoretical insights and practical methods for uncertainty quantification in various models.

Findings

Derived a representation as an infimum convolution of optimal transport and relative entropy

Provided computational methods for the divergence

Demonstrated applications in discrete and Gauss-Markov models

Abstract

This thesis develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and exploit a representation as an infimum convolution of optimal transport cost and relative entropy. We include examples of computation and approximation of the divergence, and its applications in uncertainty quantification in discrete models and Gauss-Markov models.