The goal of this paper is to complete results about I-projections and reverse I-projections, and to correct some errors in the literature. A new tool is the concept of convex support of a probability measure, better suited for our purposes than the familiar closed convex support. I. PRELIMINARIESFor probability measures (pm's) on the same measurable space, D(PIJQ) denotes information divergence (relative entropy). Its infimum for P or Q in a set S of pm's is denoted by D(SllQ) and D ( P i S ) , respectively. If here a unique minimizer exists, it is called the I-projection of Q to S or the reverse Z-projection (rI-projection) of P to S. Such projections, particularly to linear, respectively exponential families of pm's occur in various problems of probability and statistics. Previous works studying these projections include Cencov [l], Csiszk (21, [3], Topsoe [5], etc. We will consider linear families La = { P : J" x dP = a} for a E Rd, and exponential families of pm's on Rd &Q = {QB : -(z) dQB = exp[(8,z) -AQ(8)], 8 E dom AQ}, dQ where

The goal of this paper is to complete results available about-projections, reverse-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.

When finite, Shannon entropies of all subvectors of a random vector are considered for the coordinates of an entropic point in Euclidean space. A linear combination of the coordinates gives rise to an unconstrained information inequality if it is nonnegative for all entropic points. With at least four variables no finite set of linear combinations generates all such inequalities. This is proved by constructing explicitly an infinite sequence of new linear information inequalities and a curve in a special geometric position to the halfspaces defined by the inequalities. The inequalities are constructed recurrently by adhesive pasting of restrictions of polymatroids and the curve ranges in the closure of a set of the entropic points.

{ This paper presents a model of nite Radon transforms composed of Radon projections. The model generalizes to nite groups projections in the classical Radon transform theory. The Radon projector averages a function on a group over cosets of a subgroup. Reconstruction formulae formally similar to the convolved backprojection ones are derived and an iterative reconstruction technique is found to converge after nite number of steps. Applying these results to the group Z 2 p , new computationally favourable image representations have been obtained. A numerical study of the transform coding aspects is attached.

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