Aplicação da geometria diferencial à teoria da informação

Abstract

The work is based on a work due to Jacob Burbea and C. Radhakrishna Rao presented in the reference. One of the objectives of the work is the measurement of probability spaces by introducing a quadratic differential metric into the parameter space of probability distributions. For this purpose, a fi-enthalpy functional is defined on the probability space and then its Hessian along a direction of the space tangent to the space of parameters is considered as a metric in the sense of Riemannian Geometry. The distance between two probability distributions is considered to be the geodetic distance between its parameters induced by the metric. It is important to define such a distance because in drawing parameters from a probability distribution with known form we are actually a curve in a function space density and the precision in determining this function requires the existence of a measure in such a space. It was Rao himself who in 1945 noted the importance of the Differential Geometric approach. He introduced the Riemannian metric into the (differentiable) variety of a statistical model and calculated the geodesic distance between two distributions for various statistical models. These ideas have made a huge impact on the mathematical and statistical community. However, due mainly to the mathematical difficulty in this theory, it remained inert for some time. Recently properties of the Riemannian variety of a statistical model were studied by a large number of researchers independently. Even so, the statistical implications of many concepts in this geometric theory are not yet known. For example, the concept (basic in Riemannian Geometry) of curvature of a model is still devoid of statistical significance. Our work is organized as follows: In the first chapter, we present concepts necessary for the understanding of the subjects studied here. this is done in an attempt to make the work self-sufficient. In the second chapter, we present the functional fi-entropy and its hessian. Then we define the differential metric of the fi-entropy obtained from the Hessian. The chapter is finished with two examples where they are calculated at distances between two elements in their respective spaces. Finally, in the third chapter a study of the divergence measures J, K and L is made with the sole purpose of using their respective hessians to obtain the fi-entropy metric.