狄利克雷分布族的信息几何结构研究
Information Geometry of the Dirichlet Distribution Family

本文基于经典信息几何理论对狄利克雷分布族的信息几何结构进行了深入研究，首先，本文对经典信息几何的理论进行了梳理，特别是对指数分布族的信息几何结构进行了整理，给出了指数分布族统计流形在Fisher信息度量和α-联络下的几何量并给出了证明过程。其次，本文对狄利克雷分布族进行了研究，我们证明了其是一种特殊的指数分布族，构建了狄利克雷分布流形，推导出了n维狄利克雷分布流形的几何量的通项表达式。最后，计算了当狄利克雷分布为三维时，其在自然坐标系下的几何量。
This paper conducts an in-depth study on the information geometric structure of the Dirichlet distribution family based on classical information geometry theory. First, the paper systematically reviews the theoretical framework of classical information geometry, with particular focus on the geometric structure of exponential families. It provides detailed derivations and proofs of the geometric quantities for statistical manifolds of exponential families under the Fisher information metric and α-connections. Next, the research specifically addresses the Dirichlet distribution family. We prove that it constitutes a special type of exponential family, then construct the Dirichlet distribution manifold and derive general closed-form expressions for geometric quantities of n-dimensional Dirichlet manifolds. Finally, explicit computations are performed for the geometric quantities under natural coordinates in the three-dimensional case of Dirichlet distributions.