Variation and information closures of exponential families

František Matúš1

Institute of Information Th. and Automation
Academy of Sciences of the Czech Republic
Pod vodárenskou věží 4
182 08 Prague, Czech Republic

The variation distance closure of an exponential family with a convex set of canonical parameters is described, not assuming any regularity conditions. The description relies on the concept of convex core of a measure and its faces. The closure is a subset of an extension of the exponential family, defined as a union of exponential families over the faces. The crucial new ingedient is a concept of accessible faces of a convex set. The closures in reversed information divergence are expressed via the variation closures of auxiliary subfamilies. Also, variation convergence and information convergences in the extension are characterized.


joint work with Imre Csiszár