Geometrical statistics - Classical and quantum

This is a review of the ideas behind the Fisher-Rao metric on classical probability distributions, and how they generalize to metrics on density matrices. As is well known, the unique Fisher-Rao metric then becomes a large family of monotone metrics. Finally I focus on the Bures-Uhlmann metric, and discuss a recent result that connects the geometric operator mean to a geodesic billiard on the set of density matrices.