Information Geometry and Classical Cramér-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: $\alpha$-version of CR inequality, generalized CR inequality, Bayesian CR inequality, and Bayesian $\alpha$-CR inequality. These are obtained from, respectively, $I_\alpha$-divergence (or relative $\alpha$-entropy), generalized Csisz\'ar divergence, Bayesian KL divergence, and Bayesian $I_\alpha$-divergence.