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