Harmonic exponential families on homogeneous spaces

Abstract

Exponential families play an important role in the field of information geometry. By definition, there are infinitely many exponential families. However, only a small part of them are widely used. We want to give a framework to deal with these “good” families. In the light of the observation that the sample spaces of most of them are homogeneous spaces of certain Lie groups, we propose a method to construct exponential families on homogeneous spaces G/H by taking advantage of representation theory. Families obtained by this method are G-invariant exponential families. Then the following question naturally arises: are any G-invariant exponential families on G/H obtained by this method? We give an affirmative answer to this question. More precisely, any G-invariant exponential family on G/H can be realized as a subfamily of a family obtained by our method.