Generalised Exponential Families and Associated Entropy Functions

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.