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Optimality conditions for maximizers of the information divergence from an exponent family. (English) Zbl 1149.94007
Summary: The information divergence of a probability measure \(P\) from an exponential family \({\mathcal E}\) over a finite set is defined as infimum of the divergences of \(P\) from \(Q\) subject to \(Q\in{\mathcal E}\). All directional derivatives of the divergence from \({\mathcal E}\) are explicitly found. To this end, behaviour of the conjugate of a log-Laplace transform on the boundary of its domain is analysed. The first order conditions for \(P\) to be a maximizer of the divergence from \({\mathcal E}\) are presented, including new ones when \(P\) is not projectable to \({\mathcal E}\).

94A17 Measures of information, entropy
62B10 Statistical aspects of information-theoretic topics
60A10 Probabilistic measure theory
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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[1] Ay N.: An information-geometric approach to a theory of pragmatic structuring. Ann. Probab. 30 (2002), 416-436 · Zbl 1010.62007 · doi:10.1214/aop/1020107773
[2] Ay N.: Locality of Global Stochastic Interaction in Directed Acyclic Networks. Neural Computation 14 (2002), 2959-2980 · Zbl 1079.68582 · doi:10.1162/089976602760805368 · www.ingentaconnect.com
[3] Ay N., Knauf A.: Maximizing multi-information. Kybernetika 45 (2006), 517-538 · Zbl 1249.82011 · www.kybernetika.cz · eudml:33822 · arxiv:math-ph/0702002
[4] Ay N., Wennekers T.: Dynamical properties of strongly interacting Markov chains. Neural Networks 16 (2003), 1483-1497
[5] Barndorff-Nielsen O.: Information and Exponential Families in Statistical Theory. Wiley, New York 1978 · Zbl 0387.62011
[6] Brown L. D.: Fundamentals of Statistical Exponential Families. (Lecture Notes - Monograph Series 9.) Institute of Mathematical Statistics, Hayward, CA 1986 · Zbl 0685.62002
[7] Csiszár I., Matúš F.: Information projections revisited. IEEE Trans. Inform. Theory 49 (2003), 1474-1490 · Zbl 1063.94016 · doi:10.1109/TIT.2003.810633
[8] Csiszár I., Matúš F.: Closures of exponential families. Ann. Probab. 33 (2005), 582-600 · Zbl 1068.60008 · doi:10.1214/009117904000000766 · arxiv:math/0503653
[9] Csiszár I., Matúš F.: Generalized maximum likelihood estimates for exponential families. To appear in Probab. Theory Related Fields (2008) · Zbl 1133.62039 · doi:10.1007/s00440-007-0084-z
[10] Pietra S. Della, Pietra, V. Della, Lafferty J.: Inducing features of random fields. IEEE Trans. Pattern Anal. Mach. Intell. 19 (1997), 380-393
[11] Letac G.: Lectures on Natural Exponential Families and their Variance Functions. (Monografias de Matemática 50.) Instituto de Matemática Pura e Aplicada, Rio de Janeiro 1992 · Zbl 0983.62501
[12] Matúš F.: Maximization of information divergences from binary i. i.d. sequences. Proc. IPMU 2004, Perugia 2004, Vol. 2, pp. 1303-1306
[13] Matúš F., Ay N.: On maximization of the information divergence from an exponential family. Proc. WUPES’03 (J. Vejnarová, University of Economics, Prague 2003, pp. 199-204
[14] Rockafellar R. T.: Convex Analysis. Princeton University Press, Priceton, N.J. 1970 · Zbl 0193.18401
[15] Wennekers T., Ay N.: Finite state automata resulting from temporal information maximization. Theory in Biosciences 122 (2003), 5-18 · Zbl 1090.68064 · doi:10.1162/0899766054615671
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