A NOTE ON EXTENDED ARIMOTO'S ENTROPIES

摘要

Arimoto (1971) introduced, among other things, a class of entropies for probability distributions on any finite set of elements, which includes Shannon's entropy (1948) as a special case. Restricted to the set y~2 of probability distributions P = (t, 1 - t) on a set with only two elements his class of entropies is given in terms of h_a(t)={(1/(1-α)[1-t~(1/α)+(1-t)~(1/α))~α] if α ∈ (0,∞){1}) (-[t lnt + (1-t)ln(1-t)] if α=1) (min(t,1-t) if α=0) As Vajda (2009) extended a certain class of Csiszar's /-divergences, which are closely related to Arimoto's entropies to all parameters a s α∈R, the authors of this note generalised the special case of Arimoto's entropies for probability distributions P∈ y)_2 to all α∈R (De Wet and Osterreicher, 2016). It turns out that these entropies are given for negative α=-k, k ∈ (0,∞), by h_(-k)(t)=1/(1+k) (t(1-t))/([t~(1/k)+(1-t)~(1/k)]k),t∈[0,1]. In the present note their extension to probability distributions P ∈ y_n for n ≥ 2 is investigated. In addition, a comparison of Arimoto's extended class of entropies with Renyi's and Tsallis' classes, is given. For the axiomatic characterization of the latter two classes of entropies we refer to the survey paper by Csiszar (2008).