On a method to construct exponential families by representation theory

Abstract

Exponential family plays an important role in information geometry. In arXiv:1811.01394, we introduced a method to construct an exponential family $\mathcal{P}=\{p_{\theta }{\}}_{\theta \in \Theta }$ on a homogeneous space $G/H$ from a pair $(V,v_0)$. Here $V$ is a representation of $G$ and $v_0$ is an $H$-fixed vector in $V$. Then the following questions naturally arise: (Q1) when is the correspondence $\theta \mapsto p_{\theta }$ injective? (Q2) when do distinct pairs $(V,v_0)$ and $(V^{\prime },v_0^{\prime })$ generate the same family? In this paper, we answer these two questions (Theorems 1 and 2). Moreover, in Section 3, we consider the case $(G,H)=({\mathbb{R}}_{>0},\{1\})$ with a certain representation on ${\mathbb{R}}^2$. Then we see the family obtained by our method is essentially generalized inverse Gaussian distribution (GIG).

Conference paper at Geometric Science of Information 2019