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.

Footnotes

...š¹

joint work with Imre Csiszár