Cramér–Rao lower bounds arising from generalized Csiszár divergences

Abstract

We study the geometry of probability distributions with respect to a generalized family of Csiszár f -divergences. A member of this family is the relative α-entropy which is also a Rényi 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 Cramér-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 Cramér-Rao inequalities that were derived from non-information-geometric frameworks.

Conformal mirror descent with logarithmic divergences

“… 5 Note that parameterized densities similar to ( 32 ) were studied by other authors such as [ 36 ], but their motivations were orthogonal to ours. …”