指数族作为Hessian流形的等价性及具有常Hessian截面曲率的一阶指数族的分类
Equivalence on exponential families as Hessian manifolds and classification of exponential families of order 1 with constant Hessian sectional curvature
- Tokai University（东海大学）
- The University of Tokyo（东京大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在指数族上定义等价关系，证明其等价于诱导Hessian流形同构，并分类具有常Hessian截面曲率的一阶指数族，其分类与Morris的NEF-QVF分类一致。
AI中文摘要：
我们引入指数族上的一个等价关系，并证明两个指数族在此意义下等价当且仅当相应的诱导Hessian流形是同构的。此外，我们对具有常Hessian截面曲率的一阶指数族进行分类。为此，我们证明对于一阶指数族，它具有常Hessian截面曲率当且仅当与该族等价的自然指数族具有二次方差函数（NEF-QVF）。该分类本质上与Morris（1982）对NEF-QVF的分类一致。
英文摘要：
We introduce an equivalence relation on exponential families, and prove that two exponential families are equivalent in this sense if and only if the corresponding induced Hessian manifolds are isomorphic. Moreover, we classify exponential families of order 1 with constant Hessian sectional curvature. To this end, we show that for an exponential family of order 1, it has constant Hessian sectional curvature if and only if the natural exponential family equivalent to the given family has a quadratic variance function (NEF-QVF). The classification coincides with that of NEF-QVF by Morris (1982) essentially.
