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On several representations of an uncertain body of evidence. (English) Zbl 0512.94031
Fuzzy information and decision processes, 167-181 (1982).
Summary: This paper deals with various kinds of Sugeno’s fuzzy measures which, with the exception of probability measures, have been introduced recently in the literature: \(g\)-fuzzy measures, Shafer’s belief and plausibility functions, Zadeh’s possibility measures, necessity measures among others. First are recalled the existing axiomatic theories dealing with large classes of fuzzy measures: Shafer’s belief theory and the triangular norm-based approach. From both approaches emerge three remarkable families of fuzzy measures: the probability, possibility and necessity measures. Shafer’s belief and plausibility functions can be presented via a so-called basic probability assignment (which is nothing but a random set); triangular norm-based fuzzy measures can be expressed in terms of a density; the relations existing between these two representations, basic assignment and density, are investigated for the various families of introduced fuzzy measures.
The remainder of the paper is more particularly devoted to the study of the consistency relationship existing between possibilistic and probabilistic information. A distinction is made between physical and epistemic possibility. It is shown how to derive an epistemic possibility distribution from statistical evidence (i.e. a given histogram) in a rather natural way. The results are in agreement with the loose connections which exist, according to common sense, between the probable, the possible and the credible.
For the entire collection see Zbl 0496.00019.
Reviewer: Didier Dubois

94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
28E10 Fuzzy measure theory
28B10 Group- or semigroup-valued set functions, measures and integrals