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 Cramer and
Rao becomes an equality in the case of an exponential family can be
generalised. However, this requires the introduction of escort probabilities.Comment: 20 page