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

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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.
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(This article belongs to the Special Issue Facets of Entropy - Papers presented at the workshop in Copenhagen (24-26 October 2007))