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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."

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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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