Membership Functions and Probability Measures of Fuzzy Sets

TLDR

Fuzzy sets have proven useful in control theory, pattern recognition, and medical diagnosis, yet classical probability theory has been criticized for inadequately handling uncertainties in natural language and machine learning, prompting alternatives such as possibility theory. This article argues that probability theory possesses a sufficiently rich structure to incorporate fuzzy sets within its framework. The argument rests on a subjectivistic interpretation of probability, the introduction of Laplace’s genie, and the mathematical encoding of expert testimony. Consequently, probabilities of fuzzy events can be logically induced, enabling the simultaneous treatment of multiple uncertainty types within a single problem.

Abstract

AbstractThe notion of fuzzy sets has proven useful in the context of control theory, pattern recognition, and medical diagnosis. However, it has also spawned the view that classical probability theory is unable to deal with uncertainties in natural language and machine learning, so that alternatives to probability are needed. One such alternative is what is known as “possibility theory.” Such alternatives have come into being because past attempts at making fuzzy set theory and probability theory work in concert have been unsuccessful. The purpose of this article is to develop a line of argument that demonstrates that probability theory has a sufficiently rich structure for incorporating fuzzy sets within its framework. Thus probabilities of fuzzy events can be logically induced. The philosophical underpinnings that make this happen are a subjectivistic interpretation of probability, an introduction of Laplace's famous genie, and the mathematics of encoding expert testimony. The benefit of making probability theory work in concert with fuzzy set theory is an ability to deal with different kinds of uncertainties that may arise within the same problem.KEY WORDS : Decision makingExpert testimonyFuzzy controlLaplace's genieLikelihoodMachine learningMembership functionsSubjective probability