Entropies from f-Divergences

Measuring the homogeneity or diversity of a distribution is critical in many fields, from evaluating the code produced by a cryptographic algorithm to the biodiversity as a key indicator of ecosystem health. Different measures of homogeneity are necessary to accommodate different evaluation views in different fields and applications, in the same way as we can extract different means of a same sequence. Shannon entropy, introduced to measure the content of information of a distribution, and one-parameter entropies such as Tsallis and Rényi ones, have been frequently used as homegeneity measures. In this paper, we review first general properties of an homogeneity measure, and then, based on two different approaches by Cichocki and Amari and by Topsøe respectively, we consider entropies derived from f-divergences.