BISC: The Robert Example (Revised)

From: masoud nikravesh (nikraves@eecs.berkeley.edu)
Date: Fri Jan 11 2002 - 09:44:31 MET

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    Berkeley Initiative in Soft Computing (BISC)
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    The attached abstract "The Robert Example" is for your information and
    comments, if any. Should you like your comment to be ported to the BISC
    mailing list, please e-mail it to Dr. Nikravesh
    <nikravesh@cs.berkeley.edu> with cc to me.

    With my warm regards
    Lotfi Zadeh
    ==========================

    1-02-01

                                                      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., "It was very warm during much of the early part of
    summer." 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 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
    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 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 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: zadeh@cs.berkeley.edu. Research supported in part
    by 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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