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

We study the dually flat information geometry of the Tojo-Yoshino exponential family with has sample space the Poincar\'e upper plane and parameter space the open convex cone of $2\times 2$ symmetric positive-definite matrices. Using the framework of Eaton's maximal invariant, we prove that all $f$-divergences between Tojo-Yoshino Poincar\'e 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.