Short course: The geometries of statistical models

Course description

Interest in the geometrical approach to statistical inference has been growing in the last decades. Starting from the seminal works of Rao (1945) and then Amari and Chentsov in the 80s, the inter-disciplinary field of Information Geometry (IG) is now well-established with few dedicated conferences and a journal Information Geometry. The main objects under study in IG are (i) the manifold of probability distributions and statistical models therein, for instance, a curve on this manifold is a one-dimensional model, and (ii) their invariant and (Riemannian) metric structure. We will see that regular statistical models are Riemannian manifolds, with the Fisher information matrix playing the role of the metric tensor. The Fisher metric, together with a particular pair of dually coupled affine connections, gives the statistical manifold its characteristic dually flat structure, which is at the heart of IG. Using these classical results in IG we can finally understand, e.g., why the Kullback-Leibler divergence is so useful. Applications of IG are many and diverse, see for instance the conference program (and freely available videos of the talks) of IG4DS (2022), and we will try to sketch some of them.

List of topics

• Preliminaries (some concepts and results from differential geometry and geometric analysis)
• The dually flat structure of exponential families
• Non-parametric information geometry
• Applications

Course material

You can download here the lecture notes (in progress).

Schedule

• Wednesday 8 March 2023, 08:30-10:30 room 7 @ Povo 0
• Wednesday 15 March 2023, 08:30-10:30 room 7 @ Povo 0
• Wednesday 22 March 2023, 08:30-10:30 room 7 @ Povo 0
• Wednesday 29 March 2023, 08:30-10:30 room 7 @ Povo 0

References

• Amari, S. I., & Nagaoka, H. (2000). Methods of information geometry (Vol. 191). American Mathematical Society.
• Amari, S. I. (1997). Information geometry. Contemporary Mathematics, 203, 81-96.
• Ay, N., Jost, J., Vân Lê, H., & Schwachhöfer, L. (2017). Information geometry (Vol. 64). Cham: Springer.
• Pistone, G. (2019). Information geometry of the probability simplex: A short course. arXiv preprint arXiv:1911.01876.
• Pistone, G. (2013). Nonparametric information geometry. In Geometric Science of Information: First International Conference, GSI 2013, Paris, France, August 28-30, 2013. Proceedings (pp. 5-36). Springer Berlin Heidelberg.

General references on differential geometry:

• Do Carmo, M. P., & Flaherty Francis, J. (1992). Riemannian geometry (Vol. 6). Boston: Birkhäuser.
• Lang, S. (2012). Fundamentals of differential geometry (Vol. 191). Springer Science & Business Media.
• Petersen, P. (2006). Riemannian geometry (Vol. 171). New York: Springer.