Entropic Projections and Dominating Points

Modal-X, Université Paris Ouest, Bât. G, 200 av. de la République, 92000 Nanterre, France; leonard@u-paris10.fr

Received: 11 January 2008
Revised: 12 January 2009

Abstract

Entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers. They appear in many areas of applied mathematics such as statistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugate duality and functional analysis, criteria are derived for the existence of entropic projections, generalized entropic projections and dominating points. Representations of the generalized entropic projections are obtained. It is shown that they are the “measure component" of the solutions to some extended entropy minimization problem. This approach leads to new results and offers a unifying point of view. It also permits to extend previous results on the subject by removing unnecessary topological restrictions. As a by-product, new proofs of already known results are provided.