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Dually flat geometries of the deformed exponential family

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  • K.V., Harsha
  • K.S., Subrahamanian Moosath

Abstract

An exponential family is dually flat with respect to Amari’s ±1 connection. A deformed exponential family which is a generalization of the exponential family has two dually flat structures called the U-geometry and the χ-geometry. In the case of an exponential family invariant α-geometry gives the dually flat structure. But for a deformed exponential family, one need to consider generalized geometric structures other than the invariant α-geometry. The (F,G)-geometry on a statistical manifold is such a generalized geometry defined using a general embedding function F and a positive smooth function G. In this paper, we present the role of the (F,G)-geometry in the study of a deformed exponential family. We show that the dually flat U-geometry is the (F,G)-geometry for suitable choices of F and G. Further we show that the dully flat χ-geometry is the conformal flattening of the (F,G)-geometry for suitable F and G.

Suggested Citation

  • K.V., Harsha & K.S., Subrahamanian Moosath, 2015. "Dually flat geometries of the deformed exponential family," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 433(C), pages 136-147.
    • Handle: RePEc:eee:phsmap:v:433:y:2015:i:c:p:136-147
    DOI: 10.1016/j.physa.2015.03.023
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    References listed on IDEAS

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    1. G. Pistone, 2009. "κ-exponential models from the geometrical viewpoint," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 70(1), pages 29-37, July.
    2. Amari, Shun-ichi & Ohara, Atsumi & Matsuzoe, Hiroshi, 2012. "Geometry of deformed exponential families: Invariant, dually-flat and conformal geometries," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(18), pages 4308-4319.
    3. Kaniadakis, G. & Lissia, M. & Scarfone, A.M., 2004. "Deformed logarithms and entropies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 41-49.
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    Cited by:

    1. K. V. Harsha & Alladi Subramanyam, 2020. "Some information inequalities for statistical inference," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(5), pages 1237-1256, October.

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