Exponential families and affine Grassmannians

TL;DR

This work addresses the classification of discrete exponential families on a finite set by establishing a one-to-one correspondence between minimal $n$-dimensional exponential families and the affine Grassmannian $\\text{Graff}_{n}(C(\\Omega)/\\mathbb{R})$. Using a Lie group $G_{n}$ acting on pairs $(C,F)$ that define $p_{C,F}$, the authors prove the action is free and proper and that the orbit space is diffeomorphic to the Graff variety, enabling a geometric parametrization of all minimal exponential families. A key consequence is that minimal exponential families are in bijection with affine $n$-dimensional subspaces of $C(\\Omega)/\\mathbb{R}$, yielding a corollary of uniqueness when $|\\Omega|=n+1$. The approach combines reduction by stages with a group-action framework to connect exponential-family representations to affine Grassmannians, providing a canonical classification and opening routes for geometric analyses of statistical models on finite spaces.

Abstract

We establish a one-to-one correspondence between the set of minimal exponential families of dimension n defined on a finite sample space Ω and the affine Grassmannian associated to an appropriate vector space of functions.

Theorems & Definitions (22)

• Proposition 2.1: Reduction by stages
• Lemma 2.2
• proof
• Proposition 2.3
• Remark 2.4
• Definition 3.1
• Proposition 3.2
• proof
• Lemma 3.3
• proof