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Many-dimensional observables on Łukasiewicz tribe: constructions, conditioning and conditional independence
Kybernetika, Tome 41 (2005) no. 4, pp. 451-468 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Probability on collections of fuzzy sets can be developed as a generalization of the classical probability on $\sigma $-algebras of sets. A Łukasiewicz tribe is a collection of fuzzy sets which is closed under the standard fuzzy complementation and under the pointwise application of the Łukasiewicz t-norm to countably many fuzzy sets. An observable is a fuzzy set-valued mapping defined on a $\sigma $-algebra of sets and satisfying some additional properties; formally, the role of an observable is in a sense analogous to that of a random variable in classical probability theory. This article aims at studying and surveying some properties of observables on a Łukasiewicz tribe of fuzzy sets with a special focus on many-dimensional observables. Namely, the definition and basic construction techniques of observables are discussed. A method for a reasonable construction and interpretation of a joint observable is proposed. Further, the contribution contains results concerning conditioning of observables. We continue in our study [kroupaSC] of conditional independence in this framework and conclude that semi-graphoid properties are preserved.
Probability on collections of fuzzy sets can be developed as a generalization of the classical probability on $\sigma $-algebras of sets. A Łukasiewicz tribe is a collection of fuzzy sets which is closed under the standard fuzzy complementation and under the pointwise application of the Łukasiewicz t-norm to countably many fuzzy sets. An observable is a fuzzy set-valued mapping defined on a $\sigma $-algebra of sets and satisfying some additional properties; formally, the role of an observable is in a sense analogous to that of a random variable in classical probability theory. This article aims at studying and surveying some properties of observables on a Łukasiewicz tribe of fuzzy sets with a special focus on many-dimensional observables. Namely, the definition and basic construction techniques of observables are discussed. A method for a reasonable construction and interpretation of a joint observable is proposed. Further, the contribution contains results concerning conditioning of observables. We continue in our study [kroupaSC] of conditional independence in this framework and conclude that semi-graphoid properties are preserved.
Classification : 03E72, 06D35, 06D39, 60A86, 60B99
Keywords: state; observable; tribe of fuzzy sets; conditional independence 
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     title = {Many-dimensional observables on {{\L}ukasiewicz} tribe: constructions, conditioning and conditional independence},
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}
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Kroupa, Tomáš. Many-dimensional observables on Łukasiewicz tribe: constructions, conditioning and conditional independence. Kybernetika, Tome 41 (2005) no. 4, pp. 451-468. http://geodesic.mathdoc.fr/item/KYB_2005_41_4_a2/

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