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A note on extended Arimoto’s entropies. (English) Zbl 1397.62029

Summary: S. Arimoto [Inf. Control 19, 181–194 (1971; Zbl 0222.94022)] introduced, among other things, a class of entropies for probability distributions on any finite set of elements, which includes Shannon’s entropy C. E. Shannon [Bell Syst. Tech. J. 27, 379–423, 623–656 (1948; Zbl 1154.94303)] as a special case. Restricted to the set \({\mathcal P}_2\) of probability distributions \(P=(t,1-t)\) on a set with only two elements his class of entropies is given in terms of \[ h_\alpha(t)=\begin{cases} \frac{1}{1-\alpha}[1-(t^{1/\alpha}+(1-t)^{1/\alpha})^\alpha] &\text{if }\alpha\in(0,\infty)\setminus\{1\}\\ -[t\ln t+(1-t)\ln(1-t)] &\text{if }\alpha=1\\ \min(t,1-t) &\text{if }\alpha=0.\end{cases} \] As I. Vajda [Kybernetika 45, No. 6, 885–900 (2009; Zbl 1186.94421)] extended a certain class of Csiszár’s \(f\)-divergences, which are closely related to Arimoto’s entropies to all parameters \(\alpha\in\mathbb{R}\), the authors of this note generalised the special case of Arimoto’s entropies for probability distributions \(P\in{\mathcal P}_2\) to all \(\alpha\in\mathbb{R}\) T. de Wet and F. Österreicher [S. Afr. Stat. J. 50, No. 1, 43–64 (2016; Zbl 1397.62028)]. It tums out that these entropies are given for negative \(\alpha=-k\), \(k\in(0,\infty)\), by \[ h_{-k}(t)=\frac{1}{1+k}\frac{t(1-t)}{[t^{1/k}+(1-t)^{1/k}]^k},\quad t\in [0,1]. \] In the present note their extension to probability distributions \(P\in{\mathcal P}_n\) for \(n\geq 2\) is investigated. In addition, a comparison of Arimoto’s extended class of entropies with Rényi’s and Tsallis’ classes, is given. For the axiomatic characterization of the latter two classes of entropies we refer to the survey paper by I. Csiszár [Entropy 10, No. 3, 261–273 (2008; Zbl 1179.94043)].

MSC:

62B10 Statistical aspects of information-theoretic topics
94A17 Measures of information, entropy
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