Probability and Fuzzy Working in Concert—Honoring the Reliability Contributions of Nozer D. Singpurwalla

Abstract

Since Lotfi Zadeh introduced fuzzy logic and fuzzy sets, this theory characterizing the uncertainty of classification has a proven record in fields of computation and engineering. These successful applications, however, have been falsely interpreted as competition or replacement of probability theory by those in many statistical and mathematical communities. Such misconceptions are the result of a lack of understanding about types of uncertainties, and anchored attitudes clinging to the past. Nozer Singpurwalla, among other statisticians, came to the realization that probability and fuzzy set theory can and should work in concert (i.e., not in competition) to accommodate two different types of uncertainty present within a problem or system. The authors had the honor to collaborate with Nozer; those works are featured as successful applications of the probability measure of fuzzy sets in reliability where respective uncertainties of the outcome of events and of classification exist. This paper features those works which embody the use of Bayesian analysis and the subjective interpretation of probability.