Information Geometry: finite vs infinite sample space
- Giovanni Pistone (Collegio Carlo Alberto, Torino, Italy)
The scope of IG has been clearly defined in S. Amari and H. Nagaoka monograph by emphasizing the crucial importance of the dual 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.