User:Helgus/ Mathematical eventology
Mathematical eventology is a mathematical language of eventology; a new direction the probability theory; is based on the Kolmogorov axiomatics of probability theory added by two eventological principles: duality of notion of a set of random events and a random set of events and triad of notion (event, probability of event, value of event); studies eventological distributions — probability distributions of sets of events — and eventological structures of dependencies of sets of events.
Unlike probability theory, theory of random events focuses mainly on direct and regular studying of random events and their dependencies.
- Allocation of the theory of random events into an independent direction of probability theory;
- Developing mathematical eventful language (eventological distribution, set of random events, random set of events, event-terrace, set-means and so on), based on eventological principles of duality and triad, including
- crisp mathematical eventology (theory of random events) that studies random events and sets of random events, its probability distributions and structures of its dependencies;
- fuzzy mathematical eventology (theory of fuzzy events) that studies fuzzy events and sets of fuzzy events, its probability distributions and structures of its dependencies and generalizes fuzzy set theory[1][2][3][4][5], possibility theory[6] and Dempster-Shafer theory of evidence[7]; and also
- efficiency of theory of random events in many applied areas which is direct consequence of universality of mathematical event language
— can be considered as the basic results of mathematical eventology.
Major terms and fields of mathematical eventology
[edit]- Eventological triad: (event, probability and value)
- Probability of event
- Value of event
- Event, probability and value
- Conditional event, conditional probability and conditional value
- Value, information and entropy of event
- Conditional value, conditional information and conditional entropy
- Gibbsean eventological model "probability of event — value of event"
- Eventological duality (between sets of events and random sets of events)
- Random set of events (random event set)
- Set of random events
- Eventological distribution of a set of random events
- Set-means of a random set of events
- Set-means of a set of random events
- Eventological theory of dependencies of events
- Structures of dependencies of a set of events
- Frechet's covariances and correlations of events
- Eventological copula
- Eventological Bayes's theorem
- Eventological Mobius inversing
- Set-formulae of Mobius inversing events-terraces
- Formulae of Mobius inversing eventological distributions
- Additive set-functions and measures
Applications of eventological theory
[edit]- Eventological theory of fuzzy events
- Eventological foundation of Kahneman and Tversky theory
- Eventological portfolio analysis
- Eventological system analysis
- Eventology of making decision
- Eventological theory of set-preferences
- Eventological foundation of economics
- Eventological scoring
- Eventological direct and inverse Markowitz's problems
- Eventological market "Marshall's Cross"
- Eventological explaination of K.Blayh's paradox in theory of preferences
At bounds of eventology
[edit]- Subjective events, subjective probability and subjective value
- Gibbsean eventological model "probability of event — value of event"
- The phantom eventological distributions
References
[edit]- ^ Blyth C.R. (1972) On Simpson's Paradox and the Sure --- Thing Principle. - Journal of the American Statistical Association, June, 67, P.367-381.
- ^ Dubois D., H.Prade (1988) Possibility theory. - New York: Plenum Press.
- Feynman R.P. (1982) Simulating physics with computers. - International Journal of Theoretical Physics, Vol. 21, nos. 6/7, 467-488.
- ^ Fr'echet M. (1935) G'en'eralisations du th'eor'eme des probabilit'es totales - Fundamenta Mathematica. - 25.
- Hajek, Alan (2003) Interpretations of Probability. - The Stanford Encyclopedia of Philosophy (Summer 2003 Edition), Edward N.Zalta (ed.)
- ^ Herrnstein R.J. (1961) Relative and Absolute strength of Response as a Function of Frequency of Reinforcement. - Journal of the Experimental Analysis of Behavior, 4, 267-272.
- ^ Kahneman D., Tversky A. (1979) Prospect theory: An analysis of decisios under risk. - Econometrica, 47, 313-327.
- ^ Lefebvre V.A. (2001) Algebra of conscience. - Kluwer Academic Publishers. Dordrecht, Boston, London.
- ^ Markowitz Harry (1952) Portfolio Selection. - The Journal of Finance. Vol. VII, No. 1, March, 77-91.
- ^ Marshall Alfred A collection of Marshall's published works
- ^ Nelsen R.B. (1999) An Introduction to Copulas. - Lecture Notes in Statistics, Springer-Verlag, New York, v.139.
- ^ Russell Bertrand (1945) A History of Western Philosophy and Its Connection with Political and Social Circumstances from the Earliest Times to the Present Day, New York: Simon and Schuster.
- ^ Russell Bertrand (1948) Human Knowledge: Its Scope and Limits, London: George Allen & Unwin.
- Schrodinger Erwin (1959) Mind and Matter. - Cambridge, at the University Press.
- ^ Shafer G. (1976). A Mathematical Theory of Evidence. – Princeton University Press.
- ^ Smith Vernon (2002) Nobel Lecture.
- ^ Stoyan D., and H. Stoyan (1994) Fractals, Random Shapes and Point Fields. - Chichester: John Wiley & Sons.
- ^ Tversky A., Kahneman D. (1992) Advances in prospect theory: cumulative representation of uncertainty. - Journal of Risk and Uncertainty, 5, 297-323.
- ^ Vickrey William Paper on the history of Vickrey auctions in stamp collecting
- ^ Zadeh L.A. (1965) Fuzzy Sets. - Information and Control. - Vol.8. - P.338-353.
- ^ Zadeh L.A. (1968) Probability Measures of Fuzzy Events. - Journal of Mathematical Analysis and Applications. - Vol.10. - P.421-427.
- ^ Zadeh L.A. (1978). Fuzzy Sets as a Basis for a Theory of Possibility. – Fuzzy Sets and Systems. - Vol.1. - P.3-28.
- ^ Zadeh L.A. (2005). Toward a Generalized Theory of Uncertainty (GTU) - An Outline. - Information sciences (to appear).
- ^ Zadeh L.A. (2005). Toward a computational theory of precisiation of meaning based on fuzzy logic - the concept of cointensive precisiation. - Proceedings of IFSA-2005 World Congress.} - Beijing: Tsinghua University Press, Springer.