Fuzzification of probabilistic objects

Papčo, Martin

doi:10.2991/eusflat.2013.10

Cited by 3 publications

(3 citation statements)

References 19 publications

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“…This subsection is devoted to relationships between IF-probability [31]- [33] and fuzzy probability [15]- [18], [28].…”

Section: Jana Havlíčkovámentioning

confidence: 99%

Real functions and the extension of generalized probability measures II

Havlíčková

2015

Tatra Mountains Mathematical Publications

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ABSTRACT. We continue our study of the extensions of generalized probability measures. First, we describe some extensions of generalized random events (represented by classes of functions with values in [0,1]) to which generalized probability measures can be extended. Second, we study products of domains of probability and describe states on such products. Third, we show that the events in IF-probability, introduced by B. Riečan, form a suitable category isomorphic to a subcategory of the category of fuzzy random events. Consequently, IF-probability can be interpreted within fuzzy probability theory. We put forward some problems related to the extensions of probability domains and hint some applications.

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“…This subsection is devoted to relationships between IF-probability [31]- [33] and fuzzy probability [15]- [18], [28].…”

Section: Jana Havlíčkovámentioning

confidence: 99%

Real functions and the extension of generalized probability measures II

Havlíčková

2015

Tatra Mountains Mathematical Publications

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“…So, probability measures on A and sequentially continuous D-homomorphisms of A into I can be identified. Since there is a one-to-one correspondence between σ-fields of sets and measurable [0, 1]-valued functions (indeed, A is the set of all {0, 1}-valued elements of M(A)) and a one-to-one correspondence between probability measures on A and probability integrals on M(A), we can consider (X, M(A),p) as a fuzzified model of (X, A, p) ( [25] [3], [17], [13]) that a fuzzy observable on M(B) into M(A) can map some B ∈ B into M(A) \ A; such fuzzy observables are called genuine.…”

Section: Ii) Let H Be a Sequentially Continuous D-homomorphisms Of M(mentioning

confidence: 99%

On D-posets of fuzzy sets

Frič

2014

Mathematica Slovaca

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ABSTRACT. D-posets of fuzzy sets constitute a natural simple mathematical structure in which relevant notions of generalized probability theory can be formalized. We present a classification of D-posets leading to a hierarchy of distinguished subcategories of D-posets related to probability and study their relationships. This contributes to a better understanding of the transition from classical probability theory to fuzzy probability theory. In particular, we describe the transition from the Boolean cogenerator {0, 1} to the fuzzy cogenerator [0,1] and prove that the generated Lukasiewicz tribes form an epireflective subcategory of the bold algebras.

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“…Nontraditional cogenerators provide nontraditional models of probability theory ( [29]). D-posets have been introduced in [19] in order to model events in quantum probability.…”

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confidence: 99%

Upgrading Probability via Fractions of Events

Frič

Papčo

2016

Communications in Mathematics

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The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.

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Fuzzification of probabilistic objects

Cited by 3 publications

References 19 publications

Real functions and the extension of generalized probability measures II

Real functions and the extension of generalized probability measures II

On D-posets of fuzzy sets

Upgrading Probability via Fractions of Events

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