Exponenciais deformadas em variedades estatísticas: caracterização e análise de famílias de entropias

Abstract

In this thesis we study generalized statistical manifolds in the context of information geometry. The set of the strictly positive probability density functions, Pµ , can be provided with a C∞- Banach manifold. One contributions is in the generalization of this construct that uses injective deformed exponential functions. More precisely, the parametrization constructed in this thesis was made using a non-injective deformed exponential function which assumes zero until a cer- tain point and from then on is strictly increasing. This construction allowed us to parameterize Pµ using a greater diversity of functions. From these non-injecting deformed exponentials it is possible to define a divergence in the parameterized neighborhood and consequently the ϕ- divergence for Pµ . This divergence brings the possibility of a geometrical investigation using metric and connections. Moreover, they are related to q-divergence, metric, and connections obtained through the q-exponential function already known in the literature. We also prove that the κ-exponential and q-exponential function can be used to generalize the R´

enyi divergence. Another contribution of this work is the introduction new of ϕ-entropies that reduce, as parti- cular cases, to already known entropies. Similar results were proved for the relative entropy. Going back to the study of the injected deformed exponential functions in the construction of the parametrization it appears a normalization function. Still in this thesis, we elucidate the study of the behavior of the normalizing function near the boundary of its domain in the case where the analyzed points are in the Musielak-Orlicz class. This case was missing to complete all possible cases regarding this class of deformed functions. Given these facts, the main contribution of this thesis is the local study of statistical manifolds, using non-injected deformed exponentials, to obtain new ϕ-entropies with properties similar to the entropy already known and investigated in information geometry. These results allow us to acquire a wider range of appropriate tools for studying problems such as signal processing, optimization, numerical analysis, among others.