We examine the role of information geometry in the context of classical Cramer-Rao (CR) type inequalities. In particular, we focus on Eguchis theory of obtaining dualistic geometric structures from a divergence function and then applying Amari-Nagoakas 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 Csiszar divergence, Bayesian KL divergence, and Bayesian $I_alpha$-divergence.
تحميل البحث