Generalized Bayesian Cramér-Rao Inequality via Information Geometry of Relative $α$-Entropy

The relative $\alpha$-entropy is the R\'enyi analog of relative entropy and arises prominently in information-theoretic problems. Recent information geometric investigations on this quantity have enabled the generalization of the Cram\'{e}r-Rao inequality, which provides a lower bound for the variance of an estimator of an escort of the underlying parametric probability distribution. However, this framework remains unexamined in the Bayesian framework. In this paper, we propose a general Riemannian metric based on relative $\alpha$-entropy to obtain a generalized Bayesian Cram\'{e}r-Rao inequality. This establishes a lower bound for the variance of an unbiased estimator for the $\alpha$-escort distribution starting from an unbiased estimator for the underlying distribution. We show that in the limiting case when the entropy order approaches unity, this framework reduces to the conventional Bayesian Cram\'{e}r-Rao inequality. Further, in the absence of priors, the same framework yields the deterministic Cram\'{e}r-Rao inequality.