Abstract
We introduce generalized notions of a divergence function and a Fisher information matrix. We propose to generalize the notion of an exponential family of models by reformulating it in terms of the Fisher information matrix. Our methods are those of information geometry. The context is general enough to include applications from outside statistics.
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References
Barndorff-Nielsen, O.E.: Information and Exponential Families in Statistical Theory. J. Wiley and Sons, New York (1978)
Naudts, J.: Estimators, escort probabilities, and phi-exponential families in statistical physics. J. Ineq. Pure Appl. Math. 5, 102 (2004)
Grünwald, P.D., Dawid, A.P.: Game Theory, Maximum Entropy, Minimum Discrepancy and Robust Bayesian Decision Theory. Ann. Stat. 32, 1367–1433 (2004)
Eguchi, S.: Information geometry and statistical pattern recognition. Sugaku Expositions (Amer. Math. Soc.) 19, 197–216 (2006); originally Sūgaku 56, 380 (2004) (in Japanese)
Naudts, J.: The q-exponential family in statistical physics. Cent. Eur. J. Phys. 7, 405–413 (2009)
Amari, S., Ohara, A.: Geometry of q-Exponential Family of Probability Distributions. Entropy 13, 1170–1185 (2011)
Pistone, G.: Marginal Polytope of a Deformed Exponential Family, arXiv:1112.5123v1
Amari, S., Nagaoka, H.: Methods of Information Geometry. Translations of Mathematical Monographs. Oxford University Press, Oxford (2000); originally in Japanese, Iwanami Shoten, Tokyo (1993)
Bregman, L.M.: The relaxation method to find the common point of convex sets and its applications to the solution of problems in convex programming. USSR Comp. Math. Math. Phys. 7, 200–217 (1967)
Csiszar, I.: I-Divergence Geometry of Probability Distributions and Minimization Problems. Ann. Prob. 3, 146–158 (1975)
Amari, S., Cichocki, A.: Information geometry of divergence functions. Bull. Pol. Acad. Sc.: Techn. Sc. 58, 183–195 (2010)
Naudts, J., Anthonis, B.: Data set models and exponential families in statistical physics and beyond. Mod. Phys. Lett. B 26, 1250062 (2012)
Topsøe, F.: Exponential Families and MaxEnt Calculations for Entropy Measures of Statistical Physics, arXiv:0710.1701
Tsallis, C.: Introduction to nonextensive statistical mechanics. Springer (2009)
Naudts, J.: Generalised Thermostatistics. Springer (2011)
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Naudts, J., Anthonis, B. (2013). The Exponential Family in Abstract Information Theory. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_28
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DOI: https://doi.org/10.1007/978-3-642-40020-9_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40019-3
Online ISBN: 978-3-642-40020-9
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