Houju Hori Jr. and Yukio Matsumoto
Published: 28 November 2016
Abstract: Subjective qualifiers of Wald’s theory of decision functions are the fuzzy events of the subsequent fuzzy set theory. Wald’s notion of subjective qualifiers involves applying integral transforms to convert states of nature into fuzzy events. Probabilities of fuzzy events and arithmetic formulas for fuzzy utility function values are readily derived from Wald’s integral transforms. We have applied Zadeh’s extension principle to Wald’s integral transforms and demonstrated that fuzzy mathematics is effective when applied to multiple subjective probability distributions conjoined by OR and AND operations. In this paper, we focus on no-data problems and construct a fuzzy Bayes’ theorem for cases in which a membership function and multiple subjective probability distributions conjoined by OR or AND operations are given. In addition, we devise a formulation for the corresponding decision making problem.
Keywords: Subjective qualifier, fuzzy event, membership function, subjective probability distribution, OR-conjunction, AND-conjunction.