The possibility theory as a mathematical model of randomness and fuzziness phenomena is considered in a variant that enables the modeling of both probabilistic randomness, including that inherent in unpredictably evolving stochastic objects whose probabilistic models cannot be empirically reconstructed and nonprobabilistic randomness (fuzziness) inherent in real physical, technical, and economical objects, human-machine and expert systems, etc. Some principal distinctions between the considered variant and the known possibility theory variants, in particular, in mathematical formalism and its relationship with probability theory, substantive interpretation, and applications exemplified by solving the problems of identification and estimation optimization, empirical reconstruction of a fuzzy model for a studied object, measurement data analysis and interpretation, etc. (in the paper "Mathematical Modeling of Randomness and Fuzziness Phenomena in Scientific Studies. II. Applications") are shown.
This paper considers some elements of the optimal fuzzy decision theory that are similar to the optimal statistical decision theory, in particular, the theory of optimal fuzzy identification and optimal fuzzy hypothesis testing, such as Neyman-Pearson statistical hypothesis testing and optimal fuzzy estimation along with a sequential fuzzy identification algorithm similar to the Wald sequential statistical criterion. Some elements of the fuzzy measuring and computing transducer theory and its applications in the problems of the analysis and interpretation of measurement experiment data are given.
This article considers applications of the formalism of subjective modeling proposed in 36], based on modeling of uncertainty reflecting unreliability of subjective information and fuzziness common of its content. A subjective model of probabilistic randomness is defined and studied. It is shown that a researcher-modeler (RM) defines a subjective model of a discrete probability space as a space with plausibility and believability that de facto turns out to be a subjective model of the class of subjectively equivalent probability spaces that model an arbitrary evolving stochastic object, and the same space with plausibility and believability is its subjective model. This enables us to empirically recover a subjective model of an evolving stochastic object accurately and using a finite number of event observations, while its probabilistic model cannot be empirically recovered. A similar connection is established between equivalence classes of plausibility and believability distributions and classes of subjectively equivalent absolutely continuous probability densities. For two versions of plausibility and believability measures, entropies of plausibility and believability distributions of the values of an uncertain element (UCE) that model RM's subjective judgments as characteristics of the information content and uncertainty of his judgments are considered. It is shown that in the first version entropies have properties that are formally similar to those of Shannon entropy but due to absence of the law of large numbers (LLN) their interpretation fundamentally differs from the interpretation of Shannon entropy. In the third version there is an analog of the LLN, and its connection to the Shannon entropy was obtained for the expected value of subjective informational content/uncertainty. A subjective model M( )=(Omega,3(Omega), P (zeta,I degrees) (center dot,center dot; ), N (zeta,I degrees) (center dot,center dot; ) of an uncertain fuzzy element is considered, and an
mathematical formalism for subjective modeling, based on modelling of uncertainty, reflecting unreliability of subjective information and fuzziness that is common for its content. The model of subjective judgments on values of an unknown parameter x a X of the model M(x) of a research object is defined by the researcher-modeler as a space(1) (X, p(X), , ) with plausibility and believability measures, where x is an uncertain element taking values in X that models researcher-modeler's uncertain propositions about an unknown x a X, measures , model modalities of a researcher-modeler's subjective judgments on the validity of each x a X: the value of determines how relatively plausible, in his opinion, the equality is, while the value of determines how the inequality should be relatively believed in. Versions of plausibility Pl and believability Bel measures and pl- and bel-integrals that inherit some traits of probabilities, psychophysics and take into account interests of researcher-modeler groups are considered. It is shown that the mathematical formalism of subjective modeling, unlike "standard" mathematical modeling, aEuro cent enables a researcher-modeler to model both precise formalized knowledge and non-formalized unreliable knowledge, from complete ignorance to precise knowledge of the model of a research object, to calculate relative plausibilities and believabilities of any features of a research object that are specified by its subjective model , and if the data on observations of a research object is available, then it: aEuro cent enables him to estimate the adequacy of subjective model to the research objective, to correct it by combining subjective ideas and the observation data after testing their consistency, and, finally, to empirically recover the model of a research object.