Entropic Projections and Dominating Points

• Published in: Probability and statistics Vol. 14; pp. 343 - 381
• Main Author: Léonard, Christian
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.01.2010
• Subjects: 46N10; 60F10; 60F99; 60G57; Applied mathematics; Conditional laws of large numbers; Convex analysis; convex optimization; dominating points; entropic projections; Entropy; Functional Analysis; Information theory; Inverse problems; large deviations; Mathematical analysis; Mathematics; Numbers; Optimization; Orlicz spaces; Probability; random measures; Random variables; Statistical physics; Studies; large deviations; random measures; entropy; dominating points; Conditional laws of large numbers; convex optimization; entropic projections
• ISSN: 1292-8100; 1262-3318
• DOI: 10.1051/ps/2009003

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.