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A Deformed Exponential Statistical Manifold

1
Departamento de Matemática, Universidade Regional do Cariri, Juazeiro do Norte-CE 63041-145, Brazil
2
Departamento de Ciências Naturais, Matemática e Estatística, Universidade Federal Rural do Semi-Árido, Mossoró-RN 59625-900, Brazil
3
Curso de Engenharia de Computação, Campus Sobral, Universidade Federal do Ceará, Sobral-CE 62042-280, Brazil
4
Departamento de Engenharia de Teleinformática, Universidade Federal do Ceará, Fortaleza-CE 60020-181, Brazil
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(5), 496; https://doi.org/10.3390/e21050496
Received: 19 April 2019 / Revised: 12 May 2019 / Accepted: 13 May 2019 / Published: 15 May 2019
(This article belongs to the Section Information Theory, Probability and Statistics)
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PDF [348 KB, uploaded 15 May 2019]

Abstract

Consider μ a probability measure and P μ the set of μ -equivalent strictly positive probability densities. To endow P μ with a structure of a C -Banach manifold we use the φ -connection by an open arc, where φ is a deformed exponential function which assumes zero until a certain point and from then on is strictly increasing. This deformed exponential function has as particular cases the q-deformed exponential and κ -exponential functions. Moreover, we find the tangent space of P μ at a point p, and as a consequence the tangent bundle of P μ . We define a divergence using the q-exponential function and we prove that this divergence is related to the q-divergence already known from the literature. We also show that q-exponential and κ -exponential functions can be used to generalize of Rényi divergence. View Full-Text
Keywords: deformed exponential manifold; statistical manifold; φ-family; information geometry; exponential arcs deformed exponential manifold; statistical manifold; φ-family; information geometry; exponential arcs
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Josué Vieira, F.L.; Félix de Andrade, L.H.; Facundo Vigelis, R.; Casimiro Cavalcante, C. A Deformed Exponential Statistical Manifold. Entropy 2019, 21, 496.

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