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Entropy 2008, 10(3), 131-149; doi:10.3390/entropy-e10030131
Article

Generalised Exponential Families and Associated Entropy Functions

Department of Physics, University of Antwerpen, Groenenborgerlaan 171, 2020 Antwerpen, Belgium
Received: 26 February 2008 / Revised: 1 July 2008 / Accepted: 14 July 2008 / Published: 16 July 2008
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Abstract

A generalised notion of exponential families is introduced. It is based on the variational principle, borrowed from statistical physics. It is shown that inequivalent generalised entropy functions lead to distinct generalised exponential families. The well-known result that the inequality of Cram´er and Rao becomes an equality in the case of an exponential family can be generalised. However, this requires the introduction of escort probabilities.
Keywords: generalised entropy; maximum entropy principle; variational principle; generalised exponential family; Bregman divergence; generalised Fisher information; escort probability. generalised entropy; maximum entropy principle; variational principle; generalised exponential family; Bregman divergence; generalised Fisher information; escort probability.
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Naudts, J. Generalised Exponential Families and Associated Entropy Functions. Entropy 2008, 10, 131-149.

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