Dear Hung,
Thank you for your illuminating analyses of the Valentina
example,and your comments regarding the theory of random fuzzy sets.
Your high expertise in both probability theory and fuzzy logic is in
evidence.
Regarding the Bayesian approach outlined by Aleks Jakulin, I should
like to add the following. First, the conditioning information is
perception-based and not quantifiable. Specifically, how would wrinkles,
for example, be dealt with? Furthermore, if my perception is that
Valentina is young, how would the Bayesian approach apply? If a
subjective probability distribution is associated with young, what would
be the probability distribution associated with not young? We could
apply random sets to this example but fuzzy logic would be much simpler
to use.
Clearly, the power of probabilistic methods is enhanced when we move
from point-valued discrete probability distributions to set-valued
discrete probability distributions, that is, to random sets. The power
is enhanced further when we move from random sets to random fuzzy sets,
as you do. Please note that in my 1979 paper "Fuzzy Sets and Information
Granularity," Advances in Fuzzy Set Theory and Applications, M. Gupta,
R. Ragade and R. Yager (eds.), 3-18. Amsterdam: North-Holland Publishing
Co., 1979 (available upon request), I employed random fuzzy sets to
generalize the Dempster-Shafer framework. However, moving from random
sets to random fuzzy sets is not sufficient. What has to be done is
moving from random fuzzy sets to granule-valued distributions, as
described in my paper "Generalized Theory of Uncertainty
(GTU)--Principal Concepts and Ideas," in Computational Statistics & Data
Analysis 51, 15-46, 2006. Downloadable: http://www.sciencedirect.com/ or
available upon request. The concept of a granule is more general than
the concept of a fuzzy set. A granule is characterized by a generalized
constraint. In my view, this level of generalization is needed to
enhance the power of probability theory to a point where it can deal
with the examples given in my messages. Try the following: Most Swedes
are much taller than most Italians. What is the difference in the
average height of Swedes and the average height of Italians? A solution
is given in my GTU paper.
Theory of random fuzzy sets enriches probability theory but not to a
point where it can be said, as you do, that randomness and fuzziness can
coexist in the framework of probability theory. Many other problems
remain. One of them, as is pointed out in my JSPI paper, which is cited
in my previous message, is that in dealing with imprecise probabilities
we have to deal in addition with imprecise events, imprecise functions,
imprecise relations and other imprecise dependencies. More importantly,
a basic problem is that almost all concepts in probability theory are
bivalent, e.g, events A and B are either independent or not independent,
a process is either stationary or nonstationary, an event either occurs
or does not occur, with no shades of truth allowed. In reality, these
and other concepts are not bivalent--they are a matter of degree. It may
take a long time for this to happen, but I have no doubt that eventually
it will be recognized that bivalent logic is not the right kind of logic
to serve as a foundation for probability theory.
You have made and are continuing to make important contributions to
both probability theory and fuzzy logic, and building bridges between
them. Please continue to do so.
With my warm regards,
Lotfi
--
Lotfi A. Zadeh
Professor in the Graduate School
Director, Berkeley Initiative in Soft Computing (BISC)
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