Information Geometry

Statistical inference is constructed upon a statistical model consisting of a parameterised family of probability distributions, which forms a manifold. It is important to study the geometry of the manifold. It was Professor C. R. Rao who initiated information geometry in his monumental paper published in 1945. It not only included fundamentals of statistical inference such as the Cramér–Rao theorem and Rao–Blackwell theorem but also proposed differential geometry of a manifold of probability distributions. It is a Riemannian manifold where Fisher–Rao information plays the role of the metric tensor. It took decades for the importance of the geometrical structure to be recognised. The present article reviews the structure of the manifold of probability distributions and its applications and shows how the original idea of Professor Rao has been developed and popularised in the wide sense of statistical sciences including AI, signal processing, physical sciences and others.