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A Riemannian geometry in the q-exponential Banach manifold induced by q-divergences
(SPRINGER, 2013-01-01) Loaiza, G.; Quiceno, H.R.; Loaiza, G.; Quiceno, H.R.; Universidad EAFIT. Departamento de Ciencias; Matemáticas y Aplicaciones
For the family of non-parametric q-exponential statistical models, in a former paper, written by the same authors, a differentiable Banach manifold modelled on Lebesgue spaces of real random variables has been built. In this paper, the geometry induced on this manifold is characterized by q-divergence functionals. This geometry turns out to be a generalization of the geometry given by Fisher information metric and Levi-Civita connections. Moreover, the classical Amari's a-connections appears as special case of the q-connections ?(q). The main result is the expected one, namely the zero curvature of the manifold. © 2013 Springer-Verlag.