From: owner-uai@cs.orst.edu on behalf of Lotfi A. Zadeh
[zadeh@cs.berkeley.edu]
Sent: 17 lipca 2003 02:28
To: uai@cs.orst.edu
Subject: Re: [UAI] Maximum Entropy Principle

  Dear Kathy:

    Thank you for your highly insightful, informative and constructive 
comment. Note that in my message I begin by saying that the 
side-conditions are imprecise. Then, I give an example in which 
imprecision is concretized by stipulating that "approximately a" and 
"approximately b" are defined by their membership functions, with the 
facet understanding that there are many other ways in which imprecision 
can be concretized.
    Concretization in terms of membership functions is simplest and most 
natural. If u is a real number, than the grade of membership of u in the 
fuzzy set "approximately a" is simply the subjective degree to which u 
fits your intended meaning of "approximately a." If you interpret 
"approximately a" as a probability distribution, then various problems 
arise, among which is the problem of the meaning of conjunction. 
"Approximately a" could be interpreted--much less simply--in terms of 
random sets or, equivalently, in terms of a convex combination of 
intervals, but the results would be the same as those obtained through 
the use of the machinery of fuzzy sets.
    The crux of the difficulty in my example is that there are three 
distinct goals: (a) maximize entropy; (b) satisfy the fuzzy constraint 
on the mean; and (c) satisfy the fuzzy constraint on the variance. Thus, 
what you have is a problem in multicriterion optimization. There is no 
consensus on how such  problems should be solved. That is why I raise 
the question of how can the maximum entropy principle be applied when 
the side-conditions are imprecisely defined.

                                    With warm regards,

                                                Lotfi

--
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)