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Entropy and effective support size. (English) Zbl 1135.94316
Summary: Notion of Effective size of support (Ess) of a random variable is introduced. A small set of natural requirements that a measure of Ess should satisfy is presented. The measure with prescribed properties is in a direct (exp-) relationship to the family of Rényi’s - entropies which includes also Shannon’s entropy $$H$$. Considerations of choice of the value of imply that exp($$H$$) appears to be the most appropriate measure of Ess. Entropy and Ess can be viewed thanks to their log/exp relationship as two aspects of the same thing. In Probability and Statistics the Ess aspect could appear more basic than the entropic one.

##### MSC:
 94A17 Measures of information, entropy 60A10 Probabilistic measure theory 62B10 Statistical aspects of information-theoretic topics
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##### References:
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