A Riemannian Geometry in the \(q\)-Exponential Banach Manifold Induced by \(q\)-Divergences

  • Héctor R. QuicenoEmail author
  • Gabriel I. Loaiza
  • Juan C. Arango
Part of the Signals and Communication Technology book series (SCT)


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.


Fisher Information Orlicz Space Exponential Family Bayesian Estimator Geodesic Curve 
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  1. 1.
    Amari, S.: Differential-Geometrical Methods in Statistics. Springer, New York (1985)Google Scholar
  2. 2.
    Amari, S., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society, RI: Providence (2000) (Translated from the 1993 Japanese original by Daishi Harada)Google Scholar
  3. 3.
    Amari, S., Ohara, A.: Geometry of \(q\)-exponential family of probability distributions. Entropy 13, 1170–85 (2011)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Borges, E.P.: Manifestaões dinâmicas e termodinâmicas de sistemas não-extensivos. Tese de Dutorado, Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro (2004)Google Scholar
  5. 5.
    Cena, A., Pistone, G.: Exponential statistical manifold. Ann. Inst. Stat. Math. 59, 27–56 (2006)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Dawid, A.P.: On the conceptsof sufficiency and ancillarity in the presence of nuisance parameters. J. Roy. Stat. Soc. B 37, 248–258 (1975)Google Scholar
  7. 7.
    Efron, B.: Defining the curvature of a statistical problem (with applications to second order efficiency). Ann. Stat. 3, 1189–242 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Eguchi, S.: Second order efficiency of minimum coontrast estimator in a curved exponential family. Ann. Stat. 11, 793–03 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Furuichi, S.: Fundamental properties of Tsallis relative entropy. J. Math. Phys. 45, 4868–77 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gibilisco, P., Pistone, G.: Connections on non-parametric statistical manifolds by Orlicz space geometry. Inf. Dim. Anal. Quantum Probab. Relat. Top. 1, 325–47 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Jeffreys, H.: An invariant form for the prior probability in estimation problems. Proc. Roy. Soc. A 186, 453–61 (1946)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Kadets, M.I., Kadets, V.M.: Series in Banach spaces. In: Conditional and Undconditional Convergence. Birkaaauser Verlang, Basel (1997) (Traslated for the Russian by Andrei Iacob)Google Scholar
  13. 13.
    Kulback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)CrossRefGoogle Scholar
  14. 14.
    Loaiza, G., Quiceno, H.: A Riemannian geometry in the \(q\)-exponential Banach manifold induced by \(q\)-divergences. Geometric science of information, In: Proceedings of First International Conference on GSI 2013, pp. 737–742. Springer, Paris (2013)Google Scholar
  15. 15.
    Loaiza, G., Quiceno, H.R.: A \(q\)-exponential statistical Banach manifold. J. Math. Anal. Appl. 398, 446–6 (2013)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Naudts, J.: The \(q\)-exponential family in statistical physics. J. Phys. Conf. Ser. 201, 012003 (2010)Google Scholar
  17. 17.
    Pistone, G.: k-exponential models from the geometrical viewpoint. Eur. Phys. J. B 70, 29–37 (2009)Google Scholar
  18. 18.
    Pistone, G., Rogantin, M.-P.: The exponential statistical manifold: Parameters, orthogonality and space transformations. Bernoulli 4, 721–760 (1999)Google Scholar
  19. 19.
    Pistone, G., Sempi, C.: An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Stat. 23(5), 1543–1561 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Rao, C.R: Information and accuracy attainable in estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–91 (1945)Google Scholar
  21. 21.
    Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Zhang, J.: Referential duality and representational duality on statistical manifolds. In: Proceedings of the 2nd International Symposium on Information Geometry and its Applications, pp. 58–67, Tokyo, (2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Héctor R. Quiceno
    • 1
    Email author
  • Gabriel I. Loaiza
    • 1
  • Juan C. Arango
    • 1
  1. 1.Universidad EafitMedellinColombia, Suramérica

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