Dear Jason:
Thank you for solutions to the test problems. Your solutions show
that you have a high level of expertise in standard probability theory
(PT) and the maximum entropy principle. However, in my view your
solutions support, indirectly, my contention that PT and the maximum
entropy principle do not have a capability to deal with perception-based
information. To make my point, I will focus on the tall Swedes problem.
For convenience, I will formulate a progression of versions of this
problem. In these versions, what varies is the initial dataset. The
question is the same: What is the average height of Swedes. In the
following, a* denotes "approximately a."
Version 1 (crisp). Swedes over 20 range in height from 140cm to
220cm. Let h be the height of a Swede picked at random. I am told that
the distribution of h is uniform, and am asked, "What is the average
height of Swedes?" My answer is: 180cm. If I am asked, "Are you sure?"
my answer would be "Yes."
Now, I ask you the same question but without telling you what is the
probability distribution of h. You invoke the maximum entropy principle
and in response to my question tell me that the average height is 180cm.
But then I ask you, "Jason, are you sure that the average height is
180cm? If not, I may be in serious trouble." Your answer would have to
be: "No, I am not sure." This is a fundamental flaw of the maximum
entropy principle. Furthermore, as I have pointed out in earlier
messages, the principle is not applicable when information is
perception-based.
Version 2 (fuzzy) Swedes over 20 range in height from 140cm to
220cm. Over 70* percent are taller than 170* cm. What is the average
height of Swedes over 20? A fuzzy logic solution is described in the
attachment.
Version 3 (fuzzy) Swedes over 20 range in height from 140cm to
220cm. Over 70* percent are taller than 170*cm. Less than 10* percent
are shorter than 150*cm. Less than 15* percent are taller than 200*cm.
What is the average height of Swedes over 20?
I would be very interested in your solutions to these versions, and
your answer to my question: Are you sure that the answer is correct?
What if an incorrect answer may lead to a serious loss?
With my warm regards,
Lotfi
--
Lotfi A. Zadeh
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
[EMAIL PROTECTED]
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Fax (office): (510) 642-1712
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