XXVI International Fall Workshop on Geometry and Physics

Braga, 4-7 September 2017

The basic idea of information geometry is to consider families of probability distributions as Riemannian manifolds. In this setting one has a manifold and each point of the manifold is a probability distribution. There is a unique information measure on this manifold, which was introduced by Fisher, the so-called Fisher information, or its multidimensional generalization, the Fisher matrix. Rao used the Fisher matrix as Riemannian metric first in 1945. Considering the statistical models as differentiable manifolds and the Fisher information as Riemannian metric, the statistical models became Riemannian manifolds, and on these manifolds the differential geometrical quantities have statistical meanings. This fusion of the statistical models and the Riemannian geometry is called information geometry

From mathematical point of view, quantum mechanics may be constructed as an extension of probability theory, and it is possible to generalize many concepts in probability theory to their quantum equivalents. Such extension of the classical probability theory started in 1920, forced by the mathematical description of elementary particles, and this extension nowadays called to noncommutative probability theory. The classical concept of statistical models can be generalized to the noncommutative case and this noncommutative statistical model can be endowed by Riemannian metrics. Noncommutative information geometry studies these Riemannian geometries over the noncommutative probability spaces.

The mini-course aims to introduce the basic concepts of classical information geometry and to show how these ideas can be generalized to the noncommutative case.

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