SOME GEOMETRICAL ASPECTS OF DATA ANALYSIS AND STATISTICS

Abstract

In this paper we discuss some desirable properties that a distance between probability spaces must satisfy, and from these considerations we introduce the information metric in several different approaches. Once this information metric is introduced, its applications to data analysis and statistics are explored. For instance, the classic analysis of variance methods may be obtained through this geometrical approach, and a geometrical interpretation of likelihood estimation may be given in terms of this metric, by defining in a natural way a distance between the individuals of a statistical population. Finally we consider several geometric properties of a general class of elliptic probability distributions, exhibiting the differential metric, the sectional curvatures, the Ricci tensor, the geodesic equations and, in some cases, the evaluation of the Riemannian distance, called the Rao distance, for this family of probability distributions. Some statistical hypothesis tests are also discussed.