BISC: Zadeh/The Robert Example

From: masoud nikravesh (nikraves@eecs.berkeley.edu)
Date: Wed Jan 23 2002 - 18:54:37 MET

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    Berkeley Initiative in Soft Computing (BISC)
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    12-22-01

                                                      The Robert Example

                                                          Lotfi A.Zadeh

      Abstract

      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 zadeh@cs.berkeley.edu http://www.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html

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