Computational Information Geometry Wonderland

Shannon entropy is well-known to be concave. The entropy and cross-entropy (inaccuracy) yield the notion of relative entropy. Bregman divergences can be interpreted as relative entropies for a convex generator. The negative (and hence concave) generator can thus be interpreted as an entropy measure. Bregman divergences include both extensive (eg., Shannon) and non-extensive (eg., quadratic) entropies.

However, entropies need NOT BE always concave.

For example, Tsallis entropies (Havrda-Charvat) generalizing Shannon entropy are CONVEX for q<0

This proves also that Tsallis relative entropy is not a Bregman divergence for q<0.