**Uncertainty Quantification Working Group**

April 4, 11:30 AM, CNLS Conf. Room, TA-3, Bldg. 1690

## Linking probability and fuzzy set theories using likelihoods, membership functions, and Bayes theoem

Jane M. Booker, ESA-WR, Los Alamos National Laboratory

Kimberly F. Sellers, Carnegie Mellon University

Nozer D. Singpurwalla,
The George Washington University

### Abstract

To many, the term uncertainty means an absence of knowledge. To others the term embodies multiple sources including variability and errors. Regardless of the source, a fundamental question arises in how to characterize the various kinds of uncertainty and then combine them for a given problem in light of decision making. Examples of such complex problems include computer model verification and validation and reliability prediction applications with little or sparse data.

Ever since the introduction of fuzzy set theory in 1965 by Zadeh, probability and statistics are sometimes considered inadequate for dealing with certain kinds of uncertainty (even if data are available), and probability models only a certain type of uncertainty. Since it is quite possible that different types of uncertainty can be present in the same problem, Zadeh (1995) has claimed that “probability must be used in concert with fuzzy logic to enhance its effectiveness. In this perspective, probability theory

This presentation explores how probability theory and fuzzy set theory can work together, so that uncertainty of outcomes of experiments and imprecision can be treated in a unified and coherent manner. Both the theoretical and application of a linkage between the two theories will be presented. The linkage involves the use of fuzzy membership functions, likelihoods, and the use of Bayes Theorem. An example from reliability will illustrate the two theories working in concert, but within a probability framework.

Key words: Probability Theory, Fuzzy Set Theory, Membership Functions, Bayes Theorem