A note on the conditional probability of fuzzy subsets of a continuous domain

• Published in: Fuzzy sets and systems 1998-06, Vol.96 (2), p.211-222
• Main Authors: Baldwin, J.F., Lawry, J., Martin, T.P.
• Format: Article
• Language: English
• Subjects: Analytical method; Conditional probability of fuzzy sets; Exact sciences and technology; Logic and foundations; Mass assignment; Mathematical foundations; Mathematical logic, foundations, set theory; Mathematics; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Semantic unification; Set theory
• ISSN: 0165-0114; 1872-6801
• DOI: 10.1016/S0165-0114(96)00275-8

The notion of a mass assignment corresponding to a discrete fuzzy set is generalised to include all fuzzy sets. The definition is motivated by considering a possible mechanism by which an intelligent agent could extend set theoretic operations to fuzzy sets. This same mechanism also suggests an alternative definition for the conditional probability of fuzzy sets. It is noted that this definition is probability/possibility consistent and it is shown that in the case, where the probability distribution on the domain has a density function then the conditional probability of f given g can be expressed as the probability of f calculated relative to a conditional density function dependent on g. This result is used to develop an analytical method for calculating conditional probabilities of fuzzy subsets of the reals, where the membership functions of f and g and the density function are continuous piecewise linear functions. Such a method has applications to the development of a general semantic unification in Fril.