BISC: Zadeh/The Robert Example

From: masoud nikravesh (
Date: Wed Jan 23 2002 - 18:54:37 MET

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    Berkeley Initiative in Soft Computing (BISC)


                                                      The Robert Example

                                                          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., "Most Swedes are tall." 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 to achieve
    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 time t.

         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
    it usually 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, travel time is usually about 25
    min.; if Robert leaves at about 5:30 pm, then travel time is usually
    about 30 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 my 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.

    Then it is easy to verify that P(t)>0.9 for t>6:l5 ; P(t) is between 0

    and l for 5:45 <t< 6.l5; and P(t) <0.1 for t<5:45. Thus,
    P(t) is close to unity for t>6: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 ...

    Professor in the Graduate School, Computer Science Division
    Department of Electrical Engineering and Computer Sciences
    University of California
    Berkeley, CA 94720 -1776
    Director, Berkeley Initiative in Soft Computing (BISC)

    Address: Computer Science Division University of California Berkeley, CA 94720-1776 Tel(office): (510) 642-4959 Fax(office): (510) 642-1712 Tel(home): (510) 526-2569 Fax(home): (510) 526-2433, (510) 526-5181

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