School and workshop on

Where: Baltic Institute of Mathematics (hybrid format: Gothenburg, Sweden and online)

When: August 21st-September 1st 2023

Frank Nielsen

Information geometry primarily studies the geometric structures, dissimilarities, and statistical invariance of a family of probability distributions called the statistical model. A regular parametric statistical model can be geometrically handled as a Riemannian manifold equipped with the Fisher metric tensor which induces the Fisher-Rao geodesic distance. This Riemannian structure on the Fisher-Rao manifold was later generalized by a dual structure based on pairs of torsion-free affine connections coupled to the Fisher metric: The α-geometry. This dual structure casts light on the close interaction between statistical estimators in inference (maximum likelihood) and parametric statistical models (exponential families obtained from the principle of maximum entropy), and brings into play a generalized Pythagorean theorem useful to prove uniqueness of information projections. We will illustrate applications of information geometry in statistics, information theory, computer vision and pattern recognition, and learning of neural networks. The second part of the minicourse will present recent advances in information geometry and its applications.

- Part I: Video YT:Introduction to Information Geometry

- Readings:
- The Many Faces of Information Geometry (AMS Notices 2022)
- Elementary introduction to information geometry (Entropy 2020)

- Part II: Video YT:Recent advances and applications of information geometry
Readings:

- Maximal invariant:
- On f-Divergences Between Cauchy Distributions, IEEE Transactions on Information Theory, 2023 (arXiv:2101.12459)
- A note on the f-divergences between multivariate location-scale families with either prescribed scale matrices or location parameters, arXiv:2204.10952
- Fisher-Rao distance between multivariate normal distributions
- On the f$-divergences between hyperboloid and Poincaré distributions, arxiv:2205.13984 (2022)

- Exponential families and divergences:

- Maximal invariant: