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Characterization of $$\alpha$$-entropy with preference. (English) Zbl 0653.94004
This paper deals with the characterization of $$\alpha$$-entropy with preference. In the introduction, the entropy of order $$\alpha$$ is defined after the measure of $$\alpha$$-entropy with preference is given. The next section contains the characterization of $$\alpha$$-entropy with preference done by the following four postulates: P1 (recursivity), P2 (symmetry), P3 (differentiability) and P4 (normalization). The authors demonstrate a theorem by which the measure of $$\alpha$$-entropy satisfying P1 to P4 is shown to be the same as the measure of $$\alpha$$-entropy with preference. The next section focusses on the characterization of $$\alpha$$-entropy with preference, without assuming symmetry and on the replacement of the P2 postulate (symmetry) with a weaker postulate. Everything is achieved by introducing definitions and functions which are N-cyclic throughout all over the domain and the definition of the function left expansible by which the P5 postulate (cyclic symmetry) is introduced. P2 implies P5, but the conversion is not true and an example in this respect is provided. But P1, P4 and P5 imply P2 and therefore P5 is a weaker postulate.
The paper under discussion contains clear and efficient demonstrations which can be used for teaching purposes. The immediate applicability of this paper makes it both useful and interesting.
Reviewer: P.Cotae

##### MSC:
 94A17 Measures of information, entropy 60E99 Distribution theory
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##### References:
 [1] M. Belis, S. Guiaşu: A quantitative-qualitative measure of information in cybernetics. IEEE Trans. Inform. Theory IT-14 (1968), 593-594. [2] J. Havrda, F. Charvát: Quantification method of classification processes, concept of structural $$\alpha$$-entropy. Kybernetika 3 (1967), 1, 30-35. · Zbl 0178.22401
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