Tsallis differential entropy and divergences derived from the generalized Shannon-Khinchin axioms

In discrete systems, Shannon entropy is well known to be characterized by the Shannon-Khinchin axioms. Recently, this set of axioms was generalized for Tsallis entropy, one-parameter generalization of Shannon entropy. In continuous systems, Shannon differential entropy has been introduced as a natural extension of the above Shannon entropy without using an axiomatic approach. We derive the generalized entropy function as a solution of the functional equation determined by the generalized Shannon additivity, one of the most important axiom of the generalized Shannon-Khinchin axioms for Tsallis entropy. This generalized entropy function naturally introduces Tsallis differential entropy and two Tsallis divergences. In particular, one (Csisza¿r type) of the divergences has almost the same form as the ¿-divergence in information geometry and the other the Bregman type divergence. Our results reveal that the generalized Shannon additivity representing a branch structure of a rooted tree plays an essential role in the determination of these entropies.