1. bookVolume 74 (2019): Issue 1 (December 2019)
    Real Functons, Ideals, Measurable Functions, Functional Equations
Journal Details
License
Format
Journal
First Published
12 Nov 2012
Publication timeframe
3 times per year
Languages
English
access type Open Access

Real Functions in Stochastic Dependence

Published Online: 15 Nov 2019
Page range: 17 - 34
Received: 11 Dec 2017
Journal Details
License
Format
Journal
First Published
12 Nov 2012
Publication timeframe
3 times per year
Languages
English

In a fuzzified probability theory, random events are modeled by measurable functions into [0,1] and probability measures are replaced with probability integrals. The transition from Boolean two-valued logic to Lukasiewicz multivalued logic results in an upgraded probability theory in which we define and study asymmetrical stochastic dependence/independence and conditional probability based on stochastic channels and joint experiments so that the classical constructions follow as particular cases. Elementary categorical methods enable us to put the two theories into a perspective.

Keywords

MSC 2010

[1] ADÁMEK, J.: Theory of Mathematical Structures. Reidel, Dordrecht, 1983.Search in Google Scholar

[2] BABICOVÁ, D.: Probability integral as a linearization, Tatra Mt. Math. Publ. 72 (2018), 1–15.Search in Google Scholar

[3] BARNETT, J. H.: Origins of Boolean algebra in the logic of classes: George Boole, John Venn, and C. S. Pierce, www.maa.org/publications/periodicals/convergenceSearch in Google Scholar

[4] BUGAJSKI, S.: Statistical maps I. Basic properties, Math. Slovaca 51 (2001), 321–342.Search in Google Scholar

[5] _______ Statistical maps II. Operational random variables, Math. Slovaca 51 (2001), 343–361.Search in Google Scholar

[6] CHOVANEC, F.—KÔPKA, F.: D-posets.In: Handbook of Quantum Logic and Quantum Structures: Quantum Structures (K. Engesser et al., eds.), Elsevier, Amsterdam, 2007, pp. 367–428.Search in Google Scholar

[7] CHOVANEC, F.—DROBNÁ, E.—KÔPKA, F.—NÁNÁSIOVÁ, O.: Conditional states and independence in D-posets, Soft Comput. 14 (2014), 1027–1034.Search in Google Scholar

[8] DI NOLA, A.—DVUREČENSKIJ, A.: Product MV-algebras, Multi. Val. Logic 6 (2001), 193–215.Search in Google Scholar

[9] DVUREČENSKIJ, A.—PULMANNOVÁ, S.: New Trends in Quantum Structures.In: Mathematics and its Applications, Vol. 516, Kluwer Academic Publishers, Dordrecht; Ister Science, Bratislava, 2000.Search in Google Scholar

[10] _______ Conditional probability on σ-MV-algebras, Fuzzy Sets Syst. 155 (2005), 102–118.Search in Google Scholar

[11] ELIAŠ, P.—FRIČ, R.: Factorization of observables, Internat. J. Theoret. Phys. 56 (2017), 4073–4083.Search in Google Scholar

[12] FOULIS, D. J.—BENNETT, M. K.: Effect algebras and unsharp quantum logics, Found. Phys. 24, (1994), 1331–1352.Search in Google Scholar

[13] FRANZ, U.: What is stochastic independence? In: Quantum Probab. White Noise Anal., Vol. 16: Non-Commutativity, Infnite-Dimensionality and Probability at the Crossroads (N. Obata et al., eds.), Proc. of the RIMS Workshop on Infnite Dimensional Analysis and Quantum Probability, Kyoto, Japan, 2001 World Scientifc, 2003 (arXiv:math/0206017 [math.QA]).Search in Google Scholar

[14] FRIČ, R.: Remarks on statistical maps and fuzzy (operational) random variables,Tatra Mt. Math. Publ. 30 (2005), 21–34.Search in Google Scholar

[15] _______ On D-posets of fuzzy sets, Math. Slovaca 64 (2014), 545–554.Search in Google Scholar

[16] FRIČ, R.—PAPČO, M.: On probability domains, Internat. J. Theoret. Phys. 49 (2010), 3092–3100.Search in Google Scholar

[17] _______ A categorical approach to probability, Studia Logica 94 (2010), 215–230.Search in Google Scholar

[18] _______ Fuzzification of crisp domains,Kybernetika 46 (2010), 1009–1024.Search in Google Scholar

[19] _______ On probability domains II, Internat. J. Theoret. Phys. 50 (2011), 3778–3786.Search in Google Scholar

[20] _______ On probability domains III, Internat. J. Theoret. Phys. 54 (2015), 4237–4246.Search in Google Scholar

[21] _______ Upgrading probability via fractions of events, Commun. Math. 24 (2016), 29–41.Search in Google Scholar

[22] _______ Probability: from classical to fuzzy, Fuzzy Sets Syst. 326 (2017), 106–114.Search in Google Scholar

[23] _______ On probability domains IV, Internat. J. Theoret. Phys. 56 (2017), 4084–4091.Search in Google Scholar

[24] GUDDER, S.: Fuzzy probability theory, Demonstratio Math. 31 (1998), 235–254.Search in Google Scholar

[25] JUREČKOVÁ, M.: On the conditional expectation on probability MV-algebras with product, Soft Comput. 5 (2001), 381–385.Search in Google Scholar

[26] KALINA, M.—NÁNÁSIOVÁ, O.: Conditional states and joint distributions on MV-algebras,Kybernetika 42 (2006), 129–142.Search in Google Scholar

[27] KOLMOGOROV, A. N.: Grundbegriffe der wahrscheinlichkeitsrechnung. Springer, Berlin, 1933.Search in Google Scholar

[28] KÔPKA, F.: Quasi product on Boolean D-posets, Int.J.Theor.Phys. 47 (2008), 26–35.Search in Google Scholar

[29] KÔPKA, F.—CHOVANEC, F.: D-posets, Math. Slovaca 44 (1994), 21–34.Search in Google Scholar

[30] KROUPA, T.: Many-dimensional observables on Lukasiewicz tribe: constructions, conditioning and conditional independence,Kybernetika 41 (2005), 451–468.Search in Google Scholar

[31] _______ Conditional probability on MV-algebras, Fuzzy Sets Syst. 149 (2005), 369–381.Search in Google Scholar

[32] LOÈVE M.: Probability Theory. D. Van Nostrand, Inc., Princeton, New Jersey, 1963.Search in Google Scholar

[33] MESIAR, R.: Fuzzy sets and probability theory, Tatra Mt. Math. Publ. 1 (1992), 105–123.Search in Google Scholar

[34] NAVARA, M.: Probability theory of fuzzy events. In: 4th Conference of the European Society for Fuzzy Logic and Technology and 11 Rencontres Francophones sur la Logique Floue et ses Applications (E. Montseny and P. Sobrevilla eds.), Universitat Politecnica de Catalunya, Barcelona, Spain, 2005, pp. 325–329.Search in Google Scholar

[35] PAPČO, M.: On measurable spaces and measurable maps, Tatra Mt. Math. Publ. 28 (2004), 125–140.Search in Google Scholar

[36] _______ On fuzzy random variables: examples and generalizations, Tatra Mt. Math. Publ. 30 (2005), 175–185.Search in Google Scholar

[37] _______ On effect algebras, Soft Comput. 12 (2008), 373–379.Search in Google Scholar

[38] _______ Fuzzification of probabilistic objects. In: 8th Conference of the European Society for Fuzzy Logic and Technology—EUSFLAT ‘13) (G. Pasi et al., ed.), Milan, Italy, 2013, Atlantis Press, Amsterdam, 2013, pp. 67–71, doi:10.2991/eusat.2013.10 (2013).Search in Google Scholar

[39] RIEČAN, B.: On the product MV-algebras, Tatra Mt. Math. Publ. 16 (1999), 143–149.Search in Google Scholar

[40] RIEČAN, B.—MUNDICI, D.: Probability on MV-algebras, In: Handbook of Measure Theory, Vol. II (E. Pap, ed.), North-Holland, Amsterdam, 2002, pp. 869–910.Search in Google Scholar

[41] RIEČAN, B.—NEUBRUNN, T.: Integral, Measure, and Ordering. Kluwer Acad. Publ., Dordrecht, 1997.Search in Google Scholar

[42] SKŘIVÁNEK, V.—FRIČ, R.: Generalized random events, Internat. J. Theoret. Phys. 54 (2015), 4386–4396.Search in Google Scholar

[43] VRÁBELOVÁ, M.: A note on the conditional probability on product MV-algebras,Soft Comput. 4 (2000), 58–61.Search in Google Scholar

[44] ZADEH, L. A.: Probability measures of fuzzy events, J. Math. Anal. Appl. 23 (1968), 421–27.Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo