R\'enyi-Induced Information Geometry and Hartigan's Prior Family

TL;DR

This paper explores the geometric structure induced by Re9nyi divergence, derives associated priors, and establishes their relation to Hartigan's prior family, revealing new insights into statistical manifold geometry.

Contribution

It introduces the Re9nyi-induced information geometry, derives the corresponding priors, and links them to Hartigan's prior family, highlighting their equivalence under reparameterization.

Findings

Re9nyi-geometry differs from e1-geometry in structure.

Derived the canonical Re9nyi priors as dual Re9nyi-covolumes.

Established the equivalence of Hartigan's priors with Re9nyi-priors via reparameterization.

Abstract

We derive the information geometry induced by the statistical R\'enyi divergence, namely its metric tensor, its dual parametrized connections, as well as its dual Laplacians. Based on these results, we demonstrate that the R\'enyi-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 R\'enyi-geometry, namely the dual R\'enyi-covolumes. We find that the R\'enyi-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 R\'enyi-priors ($\alpha_H = \rho$).