We study various information-theoretic measures and the information geometry of the Poincar\'e distributions and the related hyperboloid distributions, and prove that their statistical mixture models are universal density estimators of smooth densities in hyperbolic spaces. The Poincar\'e and the hyperboloid distributions are two types of hyperbolic probability distributions defined using different models of hyperbolic geometry. Namely, the Poincar\'e distributions form a triparametric bivariate exponential family whose sample space is the hyperbolic Poincar\'e upper-half plane and natural parameter space is the open 3D convex cone of two-by-two positive-definite matrices. The family of hyperboloid distributions form another exponential family which has sample space the forward sheet of the two-sheeted unit hyperboloid modeling hyperbolic geometry. In the first part, we prove that all $f$-divergences between Poincar\'e distributions can be expressed using three canonical terms using Eaton's framework of maximal group invariance. We also show that the $f$-divergences between any two Poincar\'e distributions are asymmetric except when those distributions belong to a same leaf of a particular foliation of the parameter space. We report closed-form formula for the Fisher information matrix, the Shannon's differential entropy and the Kullback-Leibler divergence. and Bhattacharyya distances between such distributions using the framework of exponential families. In the second part, we state the corresponding results for the exponential family of hyperboloid distributions by highlighting a parameter correspondence between the Poincar\'e and the hyperboloid distributions. Finally, we describe a random generator to draw variates and present two Monte Carlo methods to stochastically estimate numerically $f$-divergences between hyperbolic distributions.

翻译：我们研究了庞加莱分布及其相关双曲面分布的各种信息论度量与信息几何，并证明其统计混合模型是双曲空间中光滑密度的通用密度估计器。庞加莱分布与双曲面分布是基于双曲几何的不同模型定义的两种双曲概率分布。具体而言，庞加莱分布构成一个三元双变量指数族，其样本空间为双曲庞加莱上半平面，自然参数空间为2×2正定矩阵构成的3维开凸锥；双曲面分布族则构成另一指数族，其样本空间为建模双曲几何的双叶单位双曲面的前叶。在第一部分中，我们利用Eaton最大群不变性框架证明了庞加莱分布间的所有f-散度均可通过三个规范项表示，并证明任意两个庞加莱分布间的f-散度均不对称，除非这些分布属于参数空间某特定叶形分解的同一叶。我们给出了Fisher信息矩阵、香农微分熵、Kullback-Leibler散度及Bhattacharyya距离的闭式表达式，并基于指数族理论框架推导了分布间的相关度量。在第二部分中，通过揭示庞加莱分布与双曲面分布间的参数对应关系，我们陈述了双曲面分布指数族的相应结论。最后，我们描述了生成随机变量的采样器，并提出了两种蒙特卡洛方法以数值随机估计双曲分布间的f-散度。