Geometry of Statistical Models
This is a short course of the Mathematics for daTa scieNce study plan

Bio
Giulia Bertagnolli got her MSc (2018) and PhD (2022) in Mathematics at the University of Trento, where she is currently a postdoc at the Department of Mathematics. She is interested in the geometry of structured data (graphs, multilayer networks, functional data etc.) and in geometrical approaches to inference and statistical learning.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 interdisciplinary field of Information Geometry (IG) is now wellestablished 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 onedimensional 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 KullbackLeibler 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
 Nonparametric information geometry
 Applications
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, 8196.
 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 2830, 2013. Proceedings (pp. 536). Springer Berlin Heidelberg.
Schedule
 Wednesday 8 March 2023, 08:3010:30 room 7 @ Povo 0
 Wednesday 15 March 2023, 08:3010:30 room 7 @ Povo 0
 Wednesday 22 March 2023, 08:3010:30 room 7 @ Povo 0
 Wednesday 29 March 2023, 08:3010:30 room 7 @ Povo 0
Details
 Venue: Department of Mathematics (Povo0, room 7)
 Language: English
 The participation is free. Please send an email to Prof. Claudio Agostinelli to confirm your participation.
 For further information, please contact Prof. Claudio Agostinelli