Membership Functions and Probability Measures of Fuzzy Sets: Comment
Laviolette, Michael, Journal of the American Statistical Association
Singpurwalla and Booker (hereafter S & B) are to be commended for a novel approach to the reconciliation of fuzzy set theory with probability. In particular, I deeply appreciate their acknowledged influence of our earlier work (Laviolette, Seaman, Barrett, and Woodall 1995). Our aim in that work, inspired by Lindley (1987), was to facilitate the critical evaluation of fuzzy theory and methods by promoting greater understanding. S & B have certainly presented their work in that spirit. They have even paid us the sincerest form of flattery by reusing our opening "hook"!
Advocates have claimed for almost 40 years that fuzzy logic represents a great breakthrough as an alternative or complement to probability, but the value of a theory lies in its ability to answer previously insoluble questions. Although our comparisons were necessarily limited, we saw no evidence that fuzzy logic had passed this test in 1995 or since. Consider control mechanisms, an often-cited example of successful fuzzy applications. Although S & B dispute the interpretation of the membership function as a conditional probability, the primary contribution of Barrett and Woodall was to show that given a fuzzy controlling mechanism, one can produce an identically performing controller based on conditional probability. If one can produce results identical to fuzzy methods by other means, then the practical need for fuzzy logic is open to serious question. From a philosophical standpoint, skeptics like Haack (1979) have concluded that the case for fuzzy logic has not been made; also, Hisdal (1988a, b) has discussed membership function interpretation.
S & B cogently note that the lack of an interdisciplinary outlook among statisticians has contributed to many engineers and computer scientists search for alternatives to probability. I emphatically agree. As evidence of the insularity too prevalent in the statistical community, Hoadley and Kettenring (1990) pointed out: "What the record [of journal citations] shows is mostly statisticians citing statisticians."
The embrace of fuzzy methods by technical professionals also stands as a stark indictment of the state of statistical education. Despite years of journal discussion, conferences, and other reform efforts, the structure of the introductory engineering statistics course remains largely unchanged (see Vardeman 1991). Current courses and texts still place too much emphasis on mathematical correctness and too little emphasis on practical engineering topics like establishing specification limits (Wheeler 2003). The relatively recent introductions of exploratory data analysis and computer software only represent tinkering at the margins. To meet the needs of today's industries, statistical education needs not reformation, but transformation (Snee and Hoerl 1995, Hoerl and Snee 2002). To go back even further for excellent, but still generally ignored, advice, see Wilks (1947). Also see Deming (1975) for an excellent call to action, sadly unheeded.
S & B are to be particularly commended for their efforts to introduce operationalism into classification problems, most notably by reference to Deming (1986). …