Theory of Fuzzy Sets as a Theory of Uncertainty and Decision-Making Problems in Fuzzy Experiments

This paper develops a fuzzy-theoretic approach to uncertainty and decision-making problems. It is an alternative to the probabilistic approach though it is very similar to the latter. The paper examines fuzzy analogs of the main probabilistic concepts such as joint distribution, conditional distribution, independence, etc., and the role of these concepts in the construction of optimal strategies for fuzzy decision problems. It is shown that this approach makes it entirely possible to use the rich conceptual experience of mathematical statistics in the new context. In particular, the fuzzy variant of the Bayes principle derived in the paper plays the same role in fuzzy decision-making problems as its probabilistic prototype in the theory of statistical games. To illustrate the developed approach fuzzy point estimation problems are studied. Finally, the relation between Bayes and minimax approaches is discussed. It is demonstrated that the minimax approach is a particular case of the Bayes approach within the framework of fuzzy theories.