On musielak-orlicz function spaces and applications to information geometry

Abstract

In this thesis, Musielak–Orlicz spaces are applied to Information Geometry, where φ-families of probability distributions are constructed. Using unified notation and terminology, we collected some standard results of Musielak–Orlicz spaces. Although these spaces have been studied extensively, some questions were not answered completely. We have focused on the extension of some results and techniques to arbitrary (not necessarily finite) Musielak–Orlicz functions. In some extensions, we made use of more general formulas for the order continuous and singular components of bounded linear functionals. We found necessary and sufficient conditions for the smoothness of the Orlicz norm for arbitrary Musielak–Orlicz functions. In a φ-family, subsets of Musielak–Orlicz spaces are used as coordinate sets. We obtained φ-families by a generalization of exponential families. The exponential function found in exponential families is replaced by a φ-function. In a φ-family, the analogous of the cumulant-generating functional is a normalizing function. We defined the φ-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback–Leibler divergence.