Imre Csiszár

Alfréd Rényi Institute of Mathematics

May 16, 2008

Elements of Information Theory Workshop

Stanford University

The following problem is addressed: For a strictly convex and diﬀerentiable function γ on the open interval (0, +∞), with γ (0) deﬁned by continuity, and a measurable vector valued function φ = (φ0 , . . . , φd ) on a σ-ﬁnite measure space, with φ0 identically 1, minimize the integral of γ (g) for functions g ≥ 0 such that the integral of gφ equals a given vector a. Special cases include maximization of Shannon or Burg entropy, and minimization of Lp norm, under moment constraints. The problem will be treated in its relationship with Bregman distances, covering also cases when only a generalized minimizer exists that does not satisfy the constraints. Recent results of Csisz´ ar and Mat´ uˇs, not requiring the usual constraint qualiﬁcation (Slater condition), will also be presented. The main tools are convex duality, and the concept of convex core of a measure introduced by Csisz´ ar and Mat´ uˇs.