Information geometry and its applications

Pescara - 1-5 July 2002


Abstract

July 1 - 15.20-15.50
G.Pistone - Recent Results on Exponential Statistical Manifolds

In a paper published 1995 with C. Sempi, a definition of the manifold structure of the positive probability densities was introduced. Such manifold in modeled on Orlicz spaces with exponential Young function and is based on the representation of probabilities as non-parametric exponential models. The idea was further developed in a paper with M.-P. Rogantin (1999) with improvement of the basic contruction and a few results on the expectation parameterization on submanifolds. the theory is still lacking of important features and the basic approach, eg the use of Banach space of Orlicz type as local models in the framework of standard manifold theory has been questioned.

On the positive side, a number of new results has been derived recently and old results have been improved: it is expected that some of these improvement will be presented by the author during the meeting.

We will give a short presentation of the basic theory as we know it now, recalling what it is already known and adding the new features, expecially on the regularity of change of coordinates, cumulant function, submanifolds, alternative structures. Other important chapters, eg the theory on the tangent bundle with submanifolds and connections, or the relation with information theory will be presented by other authors. [an error occurred while processing this directive]