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

Information measures and geometry of the hyperbolic exponential families of Poincar\'e and hyperboloid distributions