Probability: From classical to fuzzy

Abstract

The fuzzification of the classical probability has been initiated by L.A. Zadeh: the classical crisp random events are extended to measurable fuzzy random events and the probability measures are extended to probability integrals. The corresponding fuzzy probability (and fuzzy statistics) can be viewed as an integral part of soft computing. We follow a different approach to fuzzification developed primarily by S. Gudder and S. Bugajski: the classical notion of random variable is modified so that new theory models both fuzzy and quantum phenomena. Our goal is to survey the second approach, GB-approach, to the fuzzification of classical probability using elementary category theory. The motivation comes primarily from quantum physics: to some crisp outcome in the sample space there corresponds a whole spectrum (a genuine probability measure) in the state space; the same situation appears in other fields of science (expert systems, fuzzy logic), too. The corresponding generalization of a random variable constitutes a stochastic channel connecting two objects. Due to the duality, the channel can be equivalently described via generalized random variables and observables. In the fuzzified probability theory, an observable can map a crisp random event to a genuine fuzzy random event. First, in a bottom-up style, we recall some constructions and point out some properties of basic notions in GB-approach. Second, in a top-down style, we outline a simple model of fuzzified probability and show that the properties suffice to build the model so that it can be seen as a minimal extension of the classical probability theory. Fuzzy random events become Łukasiewicz logic propositional functions, probability measures and other relevant maps become morphisms, and basic constructions can be described via commutative diagrams.