Information projections revisited

The goal of this paper 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 (ML) estimates for exponential families are introduced. The main tool is a new concept of extension of exponential families, based on our earlier results on convex cores of measures.