Standard Probability Theory Does not Have the Capability to Deal with Perception-based Information--the Robert Example

(Professor Lotfi A. Zadeh)

The Robert Example is named after my colleague and good friend, Robert Wilensky. The example is intended to serve as a test of the ability of standard probability theory (PT) to deal with perception-based information, e.g., "Usually Robert returns from work at about 6pm." An unorthodox view that is articulated in the following is that to add to PT the capability to process perception-based information it is necessary to generalize PT in three stages. The first stage, f-generalization, adds to PT the capability to deal with fuzzy probabilities and fuzzy events--a capability which PT lacks. The result of generalization is denoted as PT+.

The second stage, g-generalization, adds to PT+ the capability to operate on granulated (linguistic) variables and relations. Granulation plays a key role in exploiting the tolerance for imprecision for achieving robustness, tractability and data compression. G-generalization of PT+ or, equivalently, f.g-generalization of PT, is denoted as PT++.

The third stage, nl-generalization, adds to PT++ the capability to operate on information expressed in a natural language, e.g., "It is very unlikely that there will be a significant increase in the price of oil in the near future." Such information will be referred to as perception-based, and, correspondingly, nl-generalization of PT, PTp, will be referred to as perception-based probability theory. PTp subsumes PT as a special case.

The Robert Example is a relatively simple instance of problems which call for the use of PTp. Following is its description.

I want to call Robert in the evening, at a time when he is likely to be home. The question is: At what time, t, should I call Robert? The decision-relevant information is the probability, P(t), that Robert is home at about t pm.

There are three versions, in order of increasing complexity, of perception-based information which I can use to estimate P(t).

Version l. Usually Robert returns from work at about 6 pm.

Version 2. Usually Robert leaves his office at about 5:30 pm, and usually it takes about 30 minutes to get home.

Version 3. Usually Robert leaves office at about 5:30 pm. Because of traffic, travel time depends on when he leaves. Specifically, if Robert leaves at about 5:20 or earlier, then travel time is usually about 25 min.; if Robert leaves at about 5:30 pm, then travel time is usually about 30 min; if Robert leaves at 5:40 pm or later, travel time is usually about 35 min.

The problem is to compute P(t) based on this information. Using PTp, the result of computation would be a fuzzy number which represents P(t). A related problem is: What is the earliest time for which P(t) is high?

Solution of Version l using PTp is described in the paper "Toward a Perception-Based Theory of Probabilistic Reasoning with Imprecise Probabilities," which is scheduled to appear in a forthcoming issue of the Journal of Statistical Planning and Inference.

It is of interest to note that solution of a crisp version of Version l leads to counterintuitive results. Specifically, assume that with probability 0.9 Robert returns from work at 6 pm plus/minus l5 min; and that P(t) is the probability that Robert is home at exactly t pm. Then it is easy to verify that P(t)>0.9 for t>6:l5; P(t) is between 0 and l for 5:456:l5, but becomes indeterminate for t<6:l5. This phenomenon is an instance of what may be called the dilemma of "it is possible but not probable."

* 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: Research supported in part by ONR N00014-00-1-0621, ONR N00014-00-1-0621, ONR Contract N00014-99-C-0298, NASA Contract NCC2-1006, NASA Grant NAC2-117, ONR Grant N00014-96-1-0556, ONR Grant FDN0014991035, ARO Grant DAAH 04-961-0341 and the BISC Program of UC Berkeley.

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