The Fisher-Rao geometry of CES distributions
Florent Bouchard Arnaud Breloy Antoine Collas Alexandre Renaux Guillaume Ginolhac
When dealing with a parametric statistical model, a Riemannian manifold can
naturally appear by endowing the parameter space with the Fisher information
metric. The geometry induced on the parameters by this metric is then referred
to as the Fisher-Rao information geometry. Interestingly, this yields a point
of view that allows for leveragingmany tools from differential geometry. After
a brief introduction about these concepts, we will present some practical uses
of these geometric tools in the framework of elliptical distributions. This
second part of the exposition is divided into three main axes: Riemannian
optimization for covariance matrix estimation, Intrinsic Cram\'er-Rao bounds,
and classification using Riemannian distances.