Fuzzy hypotheses testing in the framework of fuzzy logic

Testing hypotheses about the probability distribution underlying the available empirical data is one of the fundamental data-analytic tasks in any application domain. Basically, it consists in checking the null hypothesis that the probability distribution, a priori assumed to belong to a certain set of distributions, actually belongs to some its narrow subset, which must be precisely delimited in advance. However, sometimes there are not enough clues for such a precise delimitation, especially if the purpose of the data analysis is explorative, a situation encountered increasingly often, due to the growing amount of routinely collected data and the increasing importance of data mining. That is why generalizations of statistical hypotheses testing to vague hypotheses have been investigated for more than a decade, so far based on the most straightforward approach—to replace the set defining the null hypothesis by a fuzzy set. In this paper, a principally different approach is presented, motivated by the observational logic and its success in automated knowledge discovery. Its key idea is to view statistical testing of a fuzzy hypothesis as the application of an appropriate generalized quantifier of a fuzzy predicate calculus to predicates describing the data. The theoretical principles of the approach are explained and its first implementations are briefly sketched.