Avanços em variedades estatísticas generalizadas: uma extensão para arcos exponencial e mistura

Abstract

The collection of all strictly positive probability densities, which are equivalent to a mea- sure μ , P μ , was endowed with a C ∞ -Banach manifold structure. This structure is based on φ -families of probability distributions. To connect two probability distributions, that is, to find a curve where the ends are two distributions and that this curve is totally con- tained in P μ , was an open question for generalized statistical manifold. In this thesis, arcs in the generalized statistical manifold are investigated. The exponential arcs and misture, already well-known in Information Geometry, can be seen as a special case of these arcs. We guarantee that the generalized misture arc is well defined. We found necessary and sufficient conditions for any two probability distributions to be connected by a generalized exponential arc, a φ -arco. We also prove that, from a deformed exponential and two fixed probability distributions, a generalization of the Rényi divergence exists, on some con- ditions this generalization of the Rényi divergence is related to φ -divergence, which can be seen as a generalization of the Kullback-Leibler divergence. The normalizing function, in a φ -family, is the analog of the cumulating generating function. We also studied the behavior of the normalization function near the boundary of the φ -family domain, which is a necessary result to the development of the generalized misture arcs, since these arcs are given from the functional ones that belong to the subdifferential of the normalizing function. Conditions were found so that generalized arcs can be taken without necessarily connecting the probability distributions to the extreme points of these arcs, that is, the arcs are opened. Another important result of this work has been to prove that the set of all the distributions connected by an open φ -arc, to a fixed probability distribution, is the φ -family of probability distributions. It is further ensured that connecting two probability distributions by open arcs is an equivalence relation for both generalized arcs.