zbMATH — the first resource for mathematics

Possibility theory. II: Conditional possibility. (English) Zbl 0955.28013
Summary: In Part II it is shown that the notion of conditional possibility can be consistently introduced in possibility theory, in very much the same way as conditional expectations and probabilities are defined in the measure- and integral-theoretic treatment of probability theory. I write down possibilistic integral equations which are formal counterparts of the integral equations used to define conditional expectations and probabilities, and use their solutions to define conditional possibilities. In all, three types of conditional possibilities, with special cases, are introduced and studied. I explain why, like conditional expectations, conditional possibilities are not uniquely defined, but can only be determined up to almost everywhere equality, and I assess the consequences of this nondeterminacy. I also show that this approach solves a number of consistency problems, extant in the literature.
See also the preceding summary of Part I and the summary of Part II below.

MSC:
 28E10 Fuzzy measure theory 03B48 Probability and inductive logic 03E72 Theory of fuzzy sets, etc.
Full Text:
References:
 [1] Burrill C. W., Measure, Integration and Probability (1972) [2] De Cooman G., Journal of Fuzzy Mathematics 2 pp 281– (1994) [3] Dubois D., In Uncertainty in Artificial Intelligence (Proceedings of the Tenth Conference (1994)) pp 195– (1994) [4] DOI: 10.1080/01969728408927749 · Zbl 0595.03016 · doi:10.1080/01969728408927749 [5] Dubois D., Thiorié des possibilités (1985) [6] Dubois. D., Possibility Theory-An Approach to Computerized Processing of Uncertainty (1988) [7] DOI: 10.1016/0888-613X(90)90007-O · Zbl 0696.03006 · doi:10.1016/0888-613X(90)90007-O [8] DOI: 10.1016/0165-0114(78)90019-2 · Zbl 0393.94050 · doi:10.1016/0165-0114(78)90019-2 [9] DOI: 10.1016/0165-0114(78)90020-9 · Zbl 0423.94052 · doi:10.1016/0165-0114(78)90020-9 [10] DOI: 10.1080/01969728908902206 · doi:10.1080/01969728908902206 [11] Shafer G., A Mathematical Theory of Evidence (1976) · Zbl 0326.62009 [12] Sugeno M., 77ic Theory of Fuzzy Integrals and Its Applications (1974) [13] DOI: 10.1016/0165-0114(78)90029-5 · Zbl 0377.04002 · doi:10.1016/0165-0114(78)90029-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.