Abstract
In this chapter we consider a deformation of the nonparametric exponential statistical models, using the Tsalli’s deformed exponentials, to construct a Banach manifold modelled on spaces of essentially bounded random variables. As a result of the construction, this manifold recovers the exponential manifold given by Pistone and Sempi up to continuous embeddings on the modeling space. The \(q\)-divergence functional plays two important roles on the manifold; on one hand, the coordinate mappings are in terms of the \(q\)-divergence functional; on the other hand, this functional induces a Riemannian geometry for which the Amari’s \(\alpha \)-connections and the Levi-Civita connections appears as special cases of the \(q\)-connections induced, \(\bigtriangledown ^{(q)}\). The main result is the flatness (zero curvature) of the manifold.
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Quiceno, H.R., Loaiza, G.I., Arango, J.C. (2014). A Riemannian Geometry in the \(q\)-Exponential Banach Manifold Induced by \(q\)-Divergences. In: Nielsen, F. (eds) Geometric Theory of Information. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-05317-2_5
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DOI: https://doi.org/10.1007/978-3-319-05317-2_5
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