Abstract

July 3 - 9.30-10.20
F.Matus, I. Csiszar - Information Projections and MLE in Exponential Families Revisited

The goal of this contribution is to complete results available about I-projections, reverse I-projections, and their generalized versions, with focus on linear and exponential families. Pythagorean-like identities and inequalities are revisited and generalized, and generalized maximum likelihood estimates for exponential families are introduced. Regularity conditions, that have been frequently imposed, can be removed. The main tool is a new concept of extension of exponential families, based on our earlier results on convex cores of measures. Given a sample from an unknown distribution in an exponential family, the maximum likelihood estimate (MLE) exists if and only if the sample mean of the directional statistic belongs to the relative interior of the domain of the convex conjugate of the cumulant generating function. We show for each point of that domain that `approximate MLEs' converge to a unique member of an information closure of the exponential family. This follows from a new refinement of Fenchel inequality. The MLE in that closure and in extensions of exponential families will be related to minimization of the information divergence in the second coordinate. [an error occurred while processing this directive]