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(2014) Geometric theory of information, Dordrecht, Springer.
A Riemannian geometry in the $$$$-exponential banach manifold induced by $$$$-divergences
Héctor R. Quiceno, Gabriel I. Loaiza, Juan C. Arango
pp. 97-117
In this chapter we consider a deformation of the nonparametric exponential statistical models, using the Tsalli's deformed exponentials, to construct a Banach manifold modelled on spaces of essentially bounded random variables. As a result of the construction, this manifold recovers the exponential manifold given by Pistone and Sempi up to continuous embeddings on the modeling space. The (q)-divergence functional plays two important roles on the manifold; on one hand, the coordinate mappings are in terms of the class="InlineEquation" id="IEq7">(q)-divergence functional; on the other hand, this functional induces a Riemannian geometry for which the Amari's (alpha )-connections and the Levi-Civita connections appears as special cases of the (q)-connections induced, (igtriangledown ^{(q)}). The main result is the flatness (zero curvature) of the manifold.
Publication details
DOI: 10.1007/978-3-319-05317-2_5
Full citation:
Quiceno, H. R. , Loaiza, G. I. , Arango, J. C. (2014)., A Riemannian geometry in the $$$$-exponential banach manifold induced by $$$$-divergences, in F. Nielsen (ed.), Geometric theory of information, Dordrecht, Springer, pp. 97-117.
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