Toward a Perception-based Theory of Probabilistic Reasoning with F-Granular Probability Distributions

(Professor Lotfi A. Zadeh)

The point of departure in perception-based probability theory, PTp, is the postulate that subjective probabilities are not numerical—as they are assumed to be in the theory of subjective probabilities. Rather, subjective probabilities are perceptions of likelihood, similar to perceptions of length, distance, speed, age, taste resemblance, etc.

Reflecting the bounded ability of human sensory organs and, ultimately, the brain, to resolve detail and store information, perceptions are intrinsically imprecise. More specifically, perceptions are f-granular in the sense that (a) the mental boundaries of perceived classes are fuzzy, and (b) the perceived values of attributes are granular, with a granule being a fuzzy clump of values drawn together by indistinguishability, similarity, proximity, or functionality. For example, perceptions of age may be described as young, middle-aged, old, very old, etc.

An immediate consequence of the postulate that subjective probabilities are perceptions of likelihood, is the postulate that subjective probabilities are f-granular. Thus, the perceived values of the subjective probability of an event may be described as low, medium, high, very high, etc.

A pivotal concept in PTp is that of a f-granular probability distribution. More specifically, if X is a real-valued random variable with granules labeled small, medium and large, then its f-granular probability distribution may be expressed symbolically as P=low\small+high\medium+low\large, with the understanding that low, for example, is a perception of the probability that X is small.

In the main, PTp is a system for computing, reasoning, and decision-making with f-granular probability distributions. Conceptually and computationally, PTp is significantly more complex than standard, measurement-based probability theory, PT. The importance of PTp derives from the fact that, as a generalization of PT, it opens the door to addressing problems and issues which are beyond the reach of PT.

* Professor in the Graduate School and Director, Berkeley initiative in Soft Computing (BISC), Computer Science Division and the Electronics Research Laboratory, Department of EECS, University of California, Berkeley, CA 94720-1776; Telephone: 510-642-4959; Fax: 510-642-1712; E-Mail:

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