Rényi-Induced Information Geometry and Hartigan's Prior Family

Abstract

We derive the information geometry induced by the statistical Rényi divergence, namely its metric tensor, its dual parametrized connections, as well as its dual Laplacians. Based on these results, we demonstrate that the Rényi-geometry, though closely related, differs in structure from Amari's well-known $\alpha$-geometry. Subsequently, we derive the canonical uniform prior distributions for a statistical manifold endowed with a Rényi-geometry, namely the dual Rényi-covolumes. We find that the Rényi-priors can be made to coincide with Takeuchi and Amari's $\alpha$-priors by a reparameterization, which is itself of particular significance in statistics. Herewith, we demonstrate that Hartigan's parametrized (${\alpha }_H$) family of priors is precisely the parametrized ($\rho$) family of Rényi-priors (${\alpha }_H=\rho$).