Probability Theory and Related Fields

, Volume 162, Issue 1–2, pp 327–364 | Cite as

Information geometry and sufficient statistics

  • Nihat Ay
  • Jürgen Jost
  • Hông Vân LêEmail author
  • Lorenz Schwachhöfer


Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari–Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. This leads to the question how the geometric structures behave under such sufficient statistics. While this is well studied in the finite sample size case, in the infinite case, we encounter technical problems concerning the appropriate topologies. Here, we introduce notions of parametrized measure models and tensor fields on them that exhibit the right behavior under statistical transformations. Within this framework, we can then handle the topological issues and show that the Fisher metric and the Amari–Chentsov tensor on statistical models in the class of symmetric 2-tensor fields and 3-tensor fields can be uniquely (up to a constant) characterized by their invariance under sufficient statistics, thereby achieving a full generalization of the original result of Chentsov to infinite sample sizes. More generally, we decompose Markov morphisms between statistical models in terms of statistics. In particular, a monotonicity result for the Fisher information naturally follows.


Fisher quadratic form Amari–Chentsov tensor Sufficient statistic Chentsov theorem 

Mathematics Subject Classification (2010)

53C99 62B05 



H.V.L. would like to thank Shun-ichi Amari for many fruitful discussions, and Giovanni Pistone for providing the articles [11, 18]. We thank Holger Bernigau for his critical helpful comments on an early version of this paper. We are grateful to the anonymous referees for their helpful remarks and suggestions. This work has been supported by the Max-Planck Institute for Mathematics in the Sciences in Leipzig, the BSI at RIKEN in Tokyo, the ASSMS, GCU in Lahore-Pakistan, the VNU for Sciences in Hanoi, the Mathematical Institute of the Academy of Sciences of the Czech Republic in Prague, and the Santa Fe Institute. We are grateful for excellent working conditions and financial support of these institutions during extended visits of some of us.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Nihat Ay
    • 1
    • 3
  • Jürgen Jost
    • 1
    • 2
    • 3
  • Hông Vân Lê
    • 4
    Email author
  • Lorenz Schwachhöfer
    • 5
  1. 1.Max-Planck-Institut für Mathematik in den NaturwissenschaftenLeipzigGermany
  2. 2.Mathematisches InstitutUniversität LeipzigLeipzigGermany
  3. 3.Santa Fe InstituteSanta FeUSA
  4. 4.Institute of Mathematics of ASCRPrague 1Czech Republic
  5. 5.Fakultät für MathematikTechnische Universität DortmundDortmundGermany

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