The Annals of Probability

Closures of exponential families

Imre Csiszár and František Matúš

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Abstract

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

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

Dates
First available in Project Euclid: 3 March 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1109868592

Digital Object Identifier
doi:10.1214/009117904000000766

Mathematical Reviews number (MathSciNet)
MR2123202

Zentralblatt MATH identifier
1068.60008

Subjects
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]

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

Citation

Csiszár, Imre; Matúš, František. Closures of exponential families. Ann. Probab. 33 (2005), no. 2, 582--600. doi:10.1214/009117904000000766. https://projecteuclid.org/euclid.aop/1109868592


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References

  • Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Wiley, New York.
  • Brown, L. D. (1986). Fundamentals of Statistical Exponential Families. IMS Lecture Notes Monogr. Ser. 9. Hayward, CA.
  • Chentsov, N. N. (1982). Statistical Decision Rules and Optimal Inference. Amer. Math. Soc., Providence, RI.
  • Csiszár, I. (1984). Sanov property, generalized \emphI-projections, and a conditional limit theorem. Ann. Probab. 12 768--793.
  • Csiszár, I. and Matúš, F. (2000). Information projections revisited. In Proc. 2000 IEEE International Symposium on Information Theory 490. IEEE, New York.
  • Csiszár, I. and Matúš, F. (2001). Convex cores of measures on $\R^d$. Studia Sci. Math. Hungar. 38 177--190.
  • Csiszár, I. and Matúš, F. (2003). Information projections revisited. IEEE Trans. Inform. Theory 49 1474--1490.
  • Csiszár, I. and Matúš, F. (2004). On information closures of exponential families: A counterexample. IEEE Trans. Inform. Theory 50 922--924.
  • Harremoës, P. (2002). The information topology. In Proc. 2002 IEEE International Symposium on Information Theory 431. IEEE, New York.
  • Hiriart-Urruty, J.-B. and Lemaréchal, C. (2001). Fundamentals of Convex Analysis. Springer, Berlin.
  • Letac, G. (1992). Lectures on Natural Exponential Families and Their Variance Functions. Monografias de Matemática 50. Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil.
  • Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press.