Information Geometry

We use the term information geometry to cover those topics concerning the use of the Fisher information matrix to define a Riemannian metric, § 2.0.5, on smooth spaces of parametric statistical models, that is, on smooth spaces of probability density functions. Amari [8, 9], Amari and Nagaoka [11], Barndorff-Nielsen and Cox [20], Kass and Vos [113] and Murray and Rice [153] provide modern accounts of the differential geometry that arises from the Fisher information metric and its relation to asymptotic inference. The Introduction by R.E. Kass in [9] provided a good summary of the background and role of information geometry in mathematical statistics. In the present monograph, we use Riemannian geometric properties of various families of probability density functions in order to obtain representations of practical situations that involve statistical models.

It has by many experts been argued that the information geometric approach may not add significantly to the understanding of the theory of parametric statistical models, and this we acknowledge. Nevertheless, we are of the opinion that there is benefit for those involved with practical modelling if essential qualitative features that are common across a wide range of applications can be presented in a way that allows geometrical tools to measure distances between and lengths along trajectories through perturbations of models of relevance. Historically, the richness of operations and structure in geometry has had a powerful influence on physics and those applications suggested new geometrical developments or methodologies; indeed, from molecular biology some years ago, the behaviour of certain enzymes in DNA manipulation led to the identification of useful geometrical operators. What we offer here is some elementary geometry to display the features common, and of most significance, to a wide range of typical statistical models for real processes. Many more geometrical tools are available to make further sophisticated studies, and we hope that these may attract the interest of those who model. For example, it would be interesting to explore the details of the role of curvature in a variety of applications, and to identify when the distinguished curves called geodesics, so important in fundamental physics, have particular significance in various real processes with essentially statistical features. Are there useful ways to compactify some parameter spaces of certain applications to benefit thereby from algebraic operations on the information geometry? Do universal connections on our information geometric spaces have a useful role in applications?