On a method to construct exponential families by representation theory
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Exponential family plays an important role in information geometry. In arXiv:1811.01394, we introduced a method to construct an exponential family P={p_θ}_θ∈Θ 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 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)=(R_>0, {1}) with a certain representation on R^2. Then we see the family obtained by our method is essentially generalized inverse Gaussian distribution (GIG).
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