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)
Address:
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