In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is a Hadamard manifold, i.e. that it is geodesically complete and has everywhere negative sectional curvature. An important consequence for applications is that the Fréchet mean of a set of Dirichlet distributions is uniquely defined in this geometry.

中文翻译：

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Dirichlet分布的Fisher-Rao几何

在本文中，我们研究了在Dirichlet分布的参数空间上由Fisher-Rao度量导出的几何。我们证明该空间是Hadamard流形，即它是大地测量学上完整的并且在任何地方都具有负截面曲率。对于应用程序而言，一个重要的结果是，在这种几何形状中唯一定义了Dirichlet分布集的Fréchet均值。