Information geometry of the Tojo-Yoshino's exponential family on the Poincaré upper plane

05/27/2022

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by   Frank Nielsen, et al.

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We study the dually flat information geometry of the Tojo-Yoshino exponential family with has sample space the Poincaré upper plane and parameter space the open convex cone of 2× 2 symmetric positive-definite matrices. Using the framework of Eaton's maximal invariant, we prove that all f-divergences between Tojo-Yoshino Poincaré distributions are functions of 3 simple determinant/trace terms. We report closed-form formula for the Fisher information matrix, the differential entropy and the Kullback-Leibler divergence and Bhattacharyya distance between such distributions.