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The likelihood interpretation as the foundation of fuzzy set theory

Cattaneo, Marco E. G. V.

Authors

Marco E. G. V. Cattaneo



Abstract

In order to use fuzzy sets in real-world applications, an interpretation for the values of membership functions is needed. The history of fuzzy set theory shows that the interpretation in terms of statistical likelihood is very natural, although the connection between likelihood and probability can be misleading. In this paper, the likelihood interpretation of fuzzy sets is reviewed: it makes fuzzy data and fuzzy inferences perfectly compatible with standard statistical analyses, and sheds some light on the central role played by extension principle and α-cuts in fuzzy set theory. Furthermore, the likelihood interpretation justifies some of the combination rules of fuzzy set theory, including the product and minimum rules for the conjunction of fuzzy sets, as well as the probabilistic-sum and bounded-sum rules for the disjunction of fuzzy sets.

Journal Article Type Article
Publication Date 2017-11
Journal International journal of approximate reasoning
Print ISSN 0888-613X
Electronic ISSN 0888-613X
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 90
Pages 333-340
APA6 Citation Cattaneo, M. E. G. V. (2017). The likelihood interpretation as the foundation of fuzzy set theory. International Journal of Approximate Reasoning, 90, (333-340). doi:10.1016/j.ijar.2017.08.006. ISSN 0888-613X
DOI https://doi.org/10.1016/j.ijar.2017.08.006
Keywords Fuzzy sets; Likelihood function; Fuzzy data; Measurement error; Fuzzy inference; Combination rules
Publisher URL http://www.sciencedirect.com/science/article/pii/S0888613X17305108
Copyright Statement ©2018, Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Additional Information This is the accepted manuscript of an article published in International journal of approximate reasoning, 2017. The version of record is available at the DOI link in this record.

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Copyright Statement
©2018, Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/





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