Cram\'er-Rao Lower Bounds Arising from Generalized Csisz\'ar Divergences

TL;DR

This paper explores the geometric structure of probability distributions using generalized Csiszár divergences, deriving a broader Cramér-Rao inequality and identifying efficient estimators within this framework.

Contribution

It introduces a generalized geometric framework based on Csiszár divergences, extending the Cramér-Rao inequality and estimator theory beyond traditional models.

Findings

Derived a generalized Fisher information metric from Csiszár divergences.

Extended the dually flat structure to new distribution classes.

Identified unbiased, efficient estimators for escort models.

Abstract

We study the geometry of probability distributions with respect to a generalized family of Csisz\'ar $f$-divergences. A member of this family is the relative $\alpha$-entropy which is also a R\'enyi analog of relative entropy in information theory and known as logarithmic or projective power divergence in statistics. We apply Eguchi's theory to derive the Fisher information metric and the dual affine connections arising from these generalized divergence functions. This enables us to arrive at a more widely applicable version of the Cram\'{e}r-Rao inequality, which provides a lower bound for the variance of an estimator for an escort of the underlying parametric probability distribution. We then extend the Amari-Nagaoka's dually flat structure of the exponential and mixer models to other distributions with respect to the aforementioned generalized metric. We show that these formulations lead us to find unbiased and efficient estimators for the escort model. Finally, we compare our work with prior results on generalized Cram\'er-Rao inequalities that were derived from non-information-geometric frameworks.