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Geometry of deformed exponential families: Invariant, dually-flat and conformal geometries

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  • Amari, Shun-ichi
  • Ohara, Atsumi
  • Matsuzoe, Hiroshi

Abstract

An information-geometrical foundation is established for the deformed exponential families of probability distributions. Two different types of geometrical structures, an invariant geometry and a flat geometry, are given to a manifold of a deformed exponential family. The two different geometries provide respective quantities such as deformed free energies, entropies and divergences. The class belonging to both the invariant and flat geometries at the same time consists of exponential and mixture families. Theq-families are characterized from the viewpoint of the invariant and flat geometries. The q-exponential family is a unique class that has the invariant and flat geometries in the extended class of positive measures. Furthermore, it is the only class of which the Riemannian metric is conformally connected with the invariant Fisher metric.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:391:y:2012:i:18:p:4308-4319
    DOI: 10.1016/j.physa.2012.04.016
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    References listed on IDEAS

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    Cited by:

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    2. Zhang, Fode & Ng, Hon Keung Tony & Shi, Yimin, 2018. "Information geometry on the curved q-exponential family with application to survival data analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 788-802.
    3. Nelson, Kenric P., 2022. "Independent Approximates enable closed-form estimation of heavy-tailed distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).
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    5. 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.
    6. 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.

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