Fuzzy P-measure Research Articles

On a possible generalization of the Bayes method of inference

The main purpose of this paper is investigation of the Bayes Formula problem for Klement's fuzzy probability measure. This measure is defined here on a family of fuzzy subsets which is, among other things, closed under complement given as an involution satisfying Zadeh's Extension Principle. Due to Ovchinnikov's result [6], we express this complement as a variable pointwise involution. Next we prove that the Bayes Formula holds for fuzzy probability measure iff it is a fuzzy P-measure, more generally defined here. Moreover, we show that there exists an automorphism between fuzzy probability space with the usual Zadeh operator of complement and fuzzy probability space with Ovchinnikov's operator of complement. So, using the Bayes method of inference, we can always replace the generalized operator of complement by Zadeh's one.

A remark on Piasecki's Bayes formula for fuzzy probability measures

Piasecki in [2, 3] has presented a theorem on the fuzzy P-measure as the unique useful fuzzy probability measure satisfying the Bayes formula. However, for some special cases this theorem fails. We show one of the possibilities how to remove this inaccuracy.

A remark on the definition of fuzzy P-measures and the Bayes formula

The notion of separated fuzzy subsets is generalized here. Using this generalization, we present a new definition of fuzzy P-measure. The equivalence of the introduced definition to a previous one is proved. In turn, the Bayes formula is extended.

Fuzzy P-measure on the real line

The fuzzy measurable space on the real line, β ϱ say, is defined by means of fuzzy relation ‘less or equal’ [5]. The main properties of β ϱ are given. These results are used for investigation of fuzzy P-measure on β ϱ . The principal emphasis is laid on the connections between a fuzzy P-measure and its cumulative distribution function. A theorem extending a fuzzy P-measure to β ϱ is shown.

On the Bayes formula for fuzzy probability measures

The fuzzy P-measure is presented here as the unique useful fuzzy probability measure satisfying the Bayes formula.