Abstract
We study notions of conditional probability and stochastic dependence/independence in an upgraded probability model in which the space of events is modeled by a full Łukasiewicz tribe of all measurable functions from some measurable space into [0, 1]. Our study is based on properties of joint experiments and the notion of stochastic channel, a construct equivalent to the notion of Markov kernel between two measurable spaces. Using the notion of a degenerated stochastic channel, a channel transmitting no stochastic information between two spaces, we define an asymmetrical independence of random experiments. Finally, we define the notion of conditional probability on full Łukasiewicz tribes.
Similar content being viewed by others
References
Babicová D (2018) Probability integral as a linearization. Tatra Mt Math Publ 72:1–15
Babicová D, Frič R (2019) Real functions in stochastic dependence. Tatra Mt Math Publ 74:47–56
Bugajski S (2001a) Statistical maps I. Basic properties. Math Slovaca 51:321–342
Bugajski S (2001b) Statistical maps II. Operational random variables. Math Slovaca 51:343–361
Chovanec F, Drobná E, Kôpka F, Nánásiová O (2014) Conditional states and independence in D-posets. Soft Comput 14:1027–1034
Di Nola A, Dvurečenskij A (2001) Product MV-algebras. Multi Val Logic 6:193–215
Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer Academic Publ. and Ister Science, Dordrecht and Bratislava
Dvurečenskij A, Pulmannová S (2005) Conditional probability on \(\sigma \)-MV-algebras. Fuzzy Sets Syst 155:102–118
Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1331–1352
Frič R (2005) Remarks on statistical maps and fuzzy (operational) random variables. Tatra Mt Math Publ 30:21–34
Frič R, Papčo M (2010a) On probability domains. Internat J Theoret Phys 49:3092–3100
Frič R, Papčo M (2010b) A categorical approach to probability. Stud Log 94:215–230
Frič R, Papčo M (2010c) Fuzzification of crisp domains. Kybernetika 46:1009–1024
Frič R, Papčo M (2011) On probability domains II. Internat J Theoret Phys 50:3778–3786
Frič R, Papčo M (2015) On probability domains III. Internat J Theoret Phys 54:4237–4246
Frič R, Papčo M (2016) Upgrading probability via fractions of events. Commun Math 24:29–41
Frič R, Papčo M (2017a) Probability: from classical to fuzzy. Fuzzy Sets Syst 326:106–114
Frič R, Papčo M (2017b) On probability domains IV. Internat J Theoret Phys 56:4084–4091
Gudder S (1998) Fuzzy probability theory. Demonstr Math 31:235–254
Jurečková M (2001) On the conditional expectation on probability MV-algebras with product. Soft Comput 5:381–385
Kalina M, Nánásiová O (2006) Conditional states and joint distributions on MV-algebras. Kybernetika 42:129–142
Kolmogorov AN (1933) Grundbegriffe der wahrscheinlichkeitsrechnung. Springer, Berlin
Kôpka F (2008) Quasi product on Boolean D-posets. Int J Theor Phys 47:26–35
Kroupa T (2005a) Many-dimensional observables on Łukasiewicz tribe: constructions, conditioning and conditional independence. Kybernetika 41:451–468
Kroupa T (2005b) Conditional probability on MV-algebras. Fuzzy Sets Syst 149:369–381
Loève M (1963) Probability theory. D. Van Nostrand Inc, Princeton
Mesiar R (1992) Fuzzy sets and probability theory. Tatra Mt Math Publ 1:105–123
Navara M (2005) Probability theory of fuzzy events. In: Montseny E, Sobrevilla P (eds.) Fourth Conference of the European society for fuzzy logic and technology (EUSFLAT 2005) and Eleventh Rencontres Francophones sur la Logique Floue et ses Applications, Barcelona, Spain, pp 325–329
Papčo M (2004) On measurable spaces and measurable maps. Tatra Mt Math Publ 28:125–140
Papčo M (2005) On fuzzy random variables: examples and generalizations. Tatra Mt Math Publ 30:175–185
Papčo M (2008) On effect algebras. Soft Comput 12:373–379
Papčo M (2013) Fuzzification of probabilistic objects. In: 8th conference of the European society for fuzzy logic and technology (EUSFLAT 2013), pp 67–71. https://doi.org/10.2991/eusat.2013.10
Riečan B (1999) On the product MV-algebras. Tatra Mt Math Publ 16:143–149
Riečan B, Mundici D (2002) Probability on MV-algebras. In: Pap E (ed) Handbook of measure theory, vol II. North-Holland, Amsterdam, pp 869–910
Vrábelová M (2000) A note on the conditional probability on product MV-algebras. Soft Comput 4:58–61
Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421–27
Acknowledgements
First author gratefully acknowledge the support by the grant of the Slovak Scientific Grant Agency VEGA No. 2/0097/20. Second author gratefully acknowledge the support by the Grant of the Slovak Research and Development Agency under contract APVV-16-0073.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Additional information
Communicated by A. Di Nola.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Eliaš, P., Frič, R. Conditional probability on full Łukasiewicz tribes. Soft Comput 24, 6521–6529 (2020). https://doi.org/10.1007/s00500-020-04762-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-020-04762-6