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This paper studies the Fisher-Rao geometry on the parameter space of betadistributions. We derive the geodesic equations and the sectional curvature,and prove that it is negative. This leads to uniqueness for the Riemanniancentroid in that space. We use this Riemannian structure to study canonicalmoments, an intrinsic representation of the moments of a probabilitydistribution. Drawing on the fact that a uniform distribution in the regularmoment space corresponds to a product of beta distributions in the canonicalmoment space, we propose a mapping from the space of canonical moments to theproduct beta manifold, allowing us to use the Fisher-Rao geometry of betadistributions to compare and analyze canonical moments.

The Fisher-Rao geometry of beta distributions applied to the study of canonical moments