A Generalized Cram\'er-Rao Bound Using Information Geometry

TL;DR

This paper extends the classical Cramér-Rao bound within information geometry by deriving a generalized bound from a new Riemannian metric based on the BHHJ divergence, with potential applications in robust estimation.

Contribution

It introduces a novel Riemannian metric derived from the BHHJ divergence and formulates a generalized Cramér-Rao bound using this metric, expanding the theoretical framework.

Findings

Derived a new Riemannian metric from BHHJ divergence.

Established a generalized Cramér-Rao bound using the new metric.

Potential applications in robust statistical estimation.

Abstract

In information geometry, statistical models are considered as differentiable manifolds, where each probability distribution represents a unique point on the manifold. A Riemannian metric can be systematically obtained from a divergence function using Eguchi's theory (1992); the well-known Fisher-Rao metric is obtained from the Kullback-Leibler (KL) divergence. The geometric derivation of the classical Cram\'er-Rao Lower Bound (CRLB) by Amari and Nagaoka (2000) is based on this metric. In this paper, we study a Riemannian metric obtained by applying Eguchi's theory to the Basu-Harris-Hjort-Jones (BHHJ) divergence (1998) and derive a generalized Cram\'er-Rao bound using Amari-Nagaoka's approach. There are potential applications for this bound in robust estimation.