Information Geometry and Classical Cram'er-Rao Type Inequalities

We examine the role of information geometry in the context of classical Cram'er-Rao (CR) type inequalities. In particular, we focus on Eguchi's theory of obtaining dualistic geometric structures from a divergence function and then applying Amari-Nagoaka's theory to obtain a CR type inequality. The classical deterministic CR inequality is derived from Kullback-Leibler (KL)-divergence. We show that this framework could be generalized to other CR type inequalities through four examples: α-version of CR inequality, generalized CR inequality, Bayesian CR inequality, and Bayesian α-CR inequality. These are obtained from, respectively, Iα-divergence (or relative α-entropy), generalized Csisz'ar divergence, Bayesian KL divergence, and Bayesian Iα-divergence.