Information geometry and classical Cramér–Rao-type inequalities

Abstract We examine the role of information geometry in the context of classical Cramer–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 Csiszar divergence, Bayesian KL divergence, and Bayesian Iα-divergence.