Information Geometry: finite vs infinite sample space

The scope of IG has been clearly defined ​in S. Amari and H. ​Nagaoka ​monograph ​by emphasizing the crucial importance of the dua​l ​affine structure of regular parametric statistical models. Amari’s research program is today still in progress e.g., the papers presented in this conference.​ ​One direction of research​ ​is​ ​the​ ​extension​ ​of the dual affine structure​ ​to non-parametric models. While the parametric assumption can ​fit​ ​most of the needs of Statististics and Machine Learning, this assumption​ ​is​ ​too restrictive​ ​in such fields as Statistical Phisics, Stochastics,​ ​or Evolution Equations. It is an easy but useful exercise to retell the geometry of finite dimensional models—-such as the probability simplex or the Gaussian distribution—-in a non-parametric language. In the first part of the talk, we give a non-parametric ​presentation of the IG of the finite probability simplex based on the idea of statistical bundle. ​After that, we discuss some of the available proposals for generalizing the same set-up to technically more complex situation.