Generalized Theory of Uncertainty: Principal Concepts and Ideas
Lotfi A. Zadeh
It is a deep-seated tradition in science to turn to probability theory when one is faced with a problem in which uncertainty plays a significant role. Underlying this tradition is the assumption that there is just one kind of uncertainty—probabilistic uncertainty. The generalized theory of uncertainty (GTU) challenges this assumption. The principal thesis of GTU is that there are many different kinds of uncertainty. Principally, there are two kinds: probabilistic uncertainty and pos- sibilistic uncertainty. In addition, there are various combinations of these uncertainties, principally probabilistic/possibilistic uncertainty and possibilistic/probabilistic uncertainty. In relation to probabilistic/ possibilistic uncertainty, a case in point is the Dempster-Shafer theory of evidence. Basically, the Dempster-Shafer theory is a theory of random sets. A random set is a probability distribution of possibility distributions.
The concept of a possibility distribution was introduced and developed in Zadeh (1995). The point of departure in this paper is the thesis that possibility is a matter of degree. There is a basic difference between the concepts of probability and possibility. The concept of probability is rooted in perception of likelihood, while the concept of possibility is rooted in perception of possibility. What is the possibility of squeezing six passengers in a five-passenger car? What is the possibility that Hans may eat four eggs for breakfast?
Sections 6.2-6.18 are a reprint of the paper by Lotfi A. Zadeh 'Generalized Theory of Uncertainty (GTU)-Principal Concepts and Ideas', Computational Statistics and Data Analysis, 51, 2006, pp. 15-46. The publishers' permission to reprint the essay is gratefully acknowledged.
Is possibility a disguised form of probability? This issue has been an object of a great deal of discussion and debate, starting with the l966 paper by Loginov in which it is suggested that a fuzzy set may be defined as a conditional probability distribution. Various links between probability and possibility are discussed in Zadeh (1995). My position has been and continues to be that probability and possibility are distinct concepts, and should be treated as such in theories of uncertainty.
The centerpiece of GTU is the concept of a generalized constraint. In GTU, information is equated to a generalized constraint. The principal generalized constraints are possibilistic, probabilistic, and veristic. As an attribute of information, a particular kind of uncertainty is associated with a particular kind of generalized constraint. In GTU, computation is governed by rules related to propagation and counter-propagation of generalized constraints.
In addition to its capacity to deal with various kinds of uncertainty, GTU has a unique capacity to compute with uncertain information described in a natural language. For this purpose, GTU employs the machinery of Computing with Words (CW)—a system of computation based on fuzzy logic. The importance of this capability derives from the fact that more often than not uncertain knowledge is described in a natural language. Simple example: X is a real-valued random variable. What we know about X is that usually X is much larger than approximately a, and usually X is much smaller than approximately b. What is the probability that X is approximately c? What is the expected value of X?