From: owner-uai@cs.orst.edu on behalf of Lotfi A. Zadeh
[Zadeh@cs.berkeley.edu]
Sent: 23 lipca 2003 18:46
To: uai@cs.orst.edu
Subject: Re:[UAI]Maximum Entropy Principle

            In comments on the maximum entropy principle, a question 
which drew attention was: What is the meaning of "approximately a?"

            Basically, there  are three ways to answer the question.

            First,  and simplest, is to interprett "apaproximately a" as 
an interval centered on  a.  The problem with this interpretation is 
that it is not a good fit to the way in which humans form  perceptions. 
 In general, perceptions do not have sharp edges, reflecting the bounded 
ability of human  sensory organs. and ultimately the brain, to resolve 
detail.

            Second, is to interpret "approximately a" as a probability 
distribution.  There are two problems  with this interpretation: (a) one 
cannot operate on the probability  distribution, i.e., form 
conjunctions, negations and  disjunctions; and (b), even if  the 
probability interpretation is accepted,  we are faced with the  problem 
of maximization under stochastic constraints -- a problem which requires 
a redefinition of the
            Third, is to interpret " approximately a" as a fuzzy set or, 
equivalently, as a possibility distribution. Alternatively, the fuzzy 
set may be interpreted as a random set or a conditional probability of  
concept  of maximum.
  "approximately a" given u, where u is a real number. In the fuzzy set 
interpretation, the grade of membership of u in the fuzzy set would be 
an answer to the question:  On the scale  from 0 to l, what is the 
degree to which u fits your perception of "approximately a." The 
corresponding question in  the conditional probability interpretation 
is: Given u, what is the probability of "approximately a?" This question 
is less natural and harder to answer than the previous question.

            In conclusion, of the three possible interpretations, the 
one that is the best fit to the way in which humans form perceptions, is 
the fuzzy set interpretation.  The fuzzy set interpretation is basically 
an elastic constraint on u.

            In his comment, Paul Snow mentioned that maximization with 
imprecise side-conditions (constraints), is treated in the l968 paper by 
Jaynes.  The paper by Jaynes is a true classic, but I could not find in 
the paper a treatment of maximization under imprecise constraints. As  
stated in my original message, the concept of maximization breaks down 
when the side-conditions are imprecise.

            Thanks for  for the constructive comments.

                                                                        
                Lotfi

       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)

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