Many-dimensional observables on Łukasiewicz tribe: constructions, conditioning and conditional independence

Tomáš Kroupa

Kybernetika (2005)

  • Volume: 41, Issue: 4, page [451]-468
  • ISSN: 0023-5954

Abstract

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Probability on collections of fuzzy sets can be developed as a generalization of the classical probability on -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 -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.

How to cite

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Kroupa, Tomáš. "Many-dimensional observables on Łukasiewicz tribe: constructions, conditioning and conditional independence." Kybernetika 41.4 (2005): [451]-468. <http://eudml.org/doc/33766>.

@article{Kroupa2005,
abstract = {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.},
author = {Kroupa, Tomáš},
journal = {Kybernetika},
keywords = {state; observable; tribe of fuzzy sets; conditional independence; state; observable; tribe of fuzzy sets; conditional independence},
language = {eng},
number = {4},
pages = {[451]-468},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Many-dimensional observables on Łukasiewicz tribe: constructions, conditioning and conditional independence},
url = {http://eudml.org/doc/33766},
volume = {41},
year = {2005},
}

TY - JOUR
AU - Kroupa, Tomáš
TI - Many-dimensional observables on Łukasiewicz tribe: constructions, conditioning and conditional independence
JO - Kybernetika
PY - 2005
PB - Institute of Information Theory and Automation AS CR
VL - 41
IS - 4
SP - [451]
EP - 468
AB - 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.
LA - eng
KW - state; observable; tribe of fuzzy sets; conditional independence; state; observable; tribe of fuzzy sets; conditional independence
UR - http://eudml.org/doc/33766
ER -

References

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