Information Geometry and Algebraic Statistics on a finite state space.

Giovanni Pistone (de Castro Statistics, Collegio Carlo Alberto, Moncalieri)

It was shown by C. R. Rao in a paper published 1945 that the set of positive probabilities on a finite state space is a Riemannian manifold1 in a way which is of interest for Statistics because the metric tensor equals the Fisher information matrix. It was later pointed out by Sun-Ichi Amari2, that it is actually possible to define two other affine geometries of Hessian type3 on top of the classical Riemannian geometry. Amari gave to this new topic the name of Information Geometry.

The term Algebraic Statistics was introduced to denote the use of Commutative Computational Algebra in Statistical Design of Experiments and in Statistical Models4. Information Geometry and Algebraic statistics are deeply connected because of the central place occupied by exponential families5 in both fields.

The present talk is a tutorial and rigorous introduction the topic. See LINK for some recent research paper.

  1. Manfredo P. do Carmo. Riemannian Geometry. Birkhäuser 1992; Serge Lang. Differential and Riemannian Manifolds. Springer 1995.

  2. The original work of Amari was published in the '80s, see Shun'ichi Amari. Differential-geometrical methods in statistics. Springer 1985. See updated presentations in: Robert E. Kass and Paul W. Vos. Geometrical Foundations of Asymptotic Inference. Wiley 1997; Shun'ichi Amari and Hiroshi Nagaoka. Methods of information geometry. AMS 2000.

  3. Hirohiko Shima. The Geometry of Hessian Structures. World Scientific 2007.

  4. The term was first used by Giovanni Pistone, Eva Riccomagno, Henry P. Wynn. Algebraic Statistics: Computational Commutative Algebra in Statistics. Chapman & Hall/CRC Monographs on Statistics & Applied Probability 2000. See the Conference Algebraic Statistics 2015, June 8-11, 2015, Department of Mathematics University of Genoa for pointers to current research.

  5. A classical monograph is by Lawrence D. Brown. Fundamentals of statistical exponential families with applications in statistical decision theory. IMS 1986. Modern algebraic devolopments are in Mateusz Michałek, Bernd Sturmfels, Caroline Uhler, Piotr Zwiernik. "Exponential Varieties." arXiv:1412.6185v1 [math.AG].