Annals of Probability

Closures of exponential families

Imre Csiszár and František Matúš

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The variation distance closure of an exponential family with a convex set of canonical parameters is described, assuming no regularity conditions. The tools are the concepts of convex core of a measure and extension of an exponential family, introduced previously by the authors, and a new concept of accessible faces of a convex set. Two other closures related to the information divergence are also characterized.

Article information

Ann. Probab., Volume 33, Number 2 (2005), 582-600.

First available in Project Euclid: 3 March 2005

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Zentralblatt MATH identifier

Primary: 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx}
Secondary: 60B10: Convergence of probability measures 62B10: Information-theoretic topics [See also 94A17] 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45]

Accessible face convex core convex support exponential family extension information divergence variation distance weak convergence


Csiszár, Imre; Matúš, František. Closures of exponential families. Ann. Probab. 33 (2005), no. 2, 582--600. doi:10.1214/009117904000000766.

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