# On a method to construct exponential families by representation theory

## TL;DR

This paper investigates a method to construct exponential families on homogeneous spaces using representation theory, answering key questions about injectivity and uniqueness, and relates the construction to the generalized inverse Gaussian distribution.

## Contribution

It provides criteria for when the constructed exponential family is injective and unique, and connects the method to known distributions like GIG.

## Key findings

- Answered when the correspondence is injective.
- Determined conditions for different pairs to generate the same family.
- Linked the construction to the generalized inverse Gaussian distribution.

## 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 $(V',v_0')$ 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).