On a method to construct exponential families by representation theory
This work provides theoretical insights into the construction of exponential families in information geometry, which is incremental for researchers in mathematical statistics and representation theory.
The paper addresses when the construction of exponential families on homogeneous spaces from representation-theoretic pairs yields injective parameter mappings and when distinct pairs generate the same family, providing answers in Theorems 1 and 2, and shows that a specific case produces the generalized inverse Gaussian distribution.
Exponential family plays an important role in information geometry. In arXiv:1811.01394, we introduced a method to construct an exponential family $\mathcal{P}=\{p_θ\}_{θ\inΘ}$ 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 $θ\mapsto p_θ$ 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).