Entropic Projections and Dominating Points

Christian Léonard

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.

Mots clés: Conditional laws of large numbers; random measures; large deviations; entropy; convex optimization; entropic projections; dominating points; Orlicz spaces

Christian Léonard. Entropic Projections and Dominating Points. ESAIM: Probability and Statistics, 2010, 14, pp.343-381. ⟨10.1051/ps/2009003⟩. ⟨hal-00654471⟩