MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Information Geometry and Sufficient Statistics

Nihat Ay, Jürgen Jost, Hông Vân Lê and Lorenz J. 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 parametrized measure models in terms of statistics. In particular, the Cram\'er-Rao inequality, a monotonicity result for the Fisher information, naturally follows.

MSC Codes:
53C99, 62B05

Related publications

2015 Repository Open Access
Nihat Ay, Jürgen Jost, Hông Vân Lê and Lorenz J. Schwachhöfer

Information geometry and sufficient statistics

In: Probability theory and related fields, 162 (2015) 1-2, pp. 327-364