Invariant Geometric Structures on Statistical Models

  • Lorenz Schwachhöfer
  • Nihat Ay
  • Jürgen Jost
  • Hông Vân Lê
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)


We review the notion of parametrized measure models and tensor fields on them, which encompasses all statistical models considered by Chentsov [6], Amari [3] and Pistone-Sempi [10]. We give a complete description of n-tensor fields that are invariant under sufficient statistics. In the cases \(n= 2\) and \(n = 3\), the only such tensors are the Fisher metric and the Amari-Chentsov tensor. While this has been shown by Chentsov [7] and Campbell [5] in the case of finite measure spaces, our approach allows to generalize these results to the cases of infinite sample spaces and arbitrary n. Furthermore, we give a generalisation of the monotonicity theorem and discuss its consequences.


Banach Space Measure Space Sample Space Regularity Assumption Logarithmic Derivative 
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  1. 1.
    Ay, N., Jost, J., Lê, H.V., Schwachhöfer, L.: Information geometry and sufficient statistics. Probab. Theor. Relat. Fields 162(1–2), 327–364 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ay, N., Jost, J., Lê, H.V., Schwachhöfer, L.: Information geometry, book in preparationGoogle Scholar
  3. 3.
    Amari, S.: Differential Geometrical Theory of Statistics. In: Amari, S.-I., Barndorff-Nielsen, O.E., Kass, R.E., Lauritzen, S.L., Rao, C.R. (eds.) Differential Geometry in Statistical Inference. Lecture Note-Monograph Series, vol. 10. Institute of Mathematical Statistics, California (1987)Google Scholar
  4. 4.
    Amari, S., Nagaoka, H.: Methods of Information Geometry. Translations of Mathematical Monographs, vol. 191. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  5. 5.
    Campbell, L.L.: An extended Chentsov characterization of a Riemannian metric. Proc. AMS 98, 135–141 (1986)zbMATHGoogle Scholar
  6. 6.
    Chentsov, N.: Category of mathematical statistics. Dokl. Acad. Nauk USSR 164, 511–514 (1965)Google Scholar
  7. 7.
    Chentsov, N.: Statistical Decision Rules and Optimal Inference. Translation of Mathematical Monograph, vol. 53. AMS, Providence (1982)zbMATHGoogle Scholar
  8. 8.
    Bauer, M., Bruveris, M., Michor, P.: Uniqueness of the Fisher-Rao metric on the space of smooth densities (2014). arXiv:1411.5577
  9. 9.
    Lang, S.: Fundamentals Of Differential Geometry. Springer, NewYork (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    Pistone, G., Sempi, C.: An infinite-dimensional structure on the space of all the probability measures equivalent to a given one. Ann. Stat. 5, 1543–1561 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rao, C.R.: Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–89 (1945)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Lorenz Schwachhöfer
    • 1
  • Nihat Ay
    • 2
  • Jürgen Jost
    • 2
  • Hông Vân Lê
    • 3
  1. 1.Technische Universität DortmundDortmundGermany
  2. 2.Max-Planck-Institut Für Mathematik in den NaturwissenschaftenLeipzigGermany
  3. 3.Mathematical Institute of ASCRPrahaCzech Republic

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