  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.
