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Fuzzy entropy and conditioning. (English) Zbl 0623.94005
A new nonprobabilistic entropy measure is introduced in the context of fuzzy sets or messages. Fuzzy units, or fits, replace bits in a new framework of fuzzy information theory. An appropriate measure of entropy or fuzziness of messages is shown to be a simple ratio of distances: the distances between the fuzzy message and its nearest and farthest nonfuzzy neighbors. Fuzzy conditioning is examined as the degree of subsethood (submessagehood) of one fuzzy set or message in another. This quantity is shown to behave as a conditional probability in many contexts. It is also shown that the entropy of A is the degree to which $$A\cup A^ c$$ is a subset of $$A\cap A^ c$$, an intuitive relationship that cannot occur in probability theory. This theory of subsethood is then shown to solve one of the major problems with Bayes-theorem learning and its variants - the problem of requiring that the space of alternatives be partitioned into disjoint exhaustive hypotheses. Any fuzzy subsets will do. However, a rough inverse relationship holds between number and fuzziness of partitions and the information gained from experience. All results reduce to fuzzy cardinality.

##### MSC:
 94A17 Measures of information, entropy 94A15 Information theory (general) 94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
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