Hello,
I have been asked to write a one-page summary on fuzzy sets --- yet another
area of which I have no knowledge --- and I therefore would be interested in
some clarification on what cases fuzzy set theory is capable of modeling
better than Bayesian Network models.
Since I am a "Bayes-man" I, of course, started to check Pearl's book, who
states that "Fuzzy logic (and sets) is orthogonal to probability theory ---
it focuses on the ambiguities in describing events, rather than the
uncertainty about the occurrence or non-occurrence of events"... and the
topic is not addressed in his book (bad luck!).
My limited source of knowledge on fuzzy sets is based primarily on
Cui, W.C (1989). "Uncertainty Analysis in Structural Safety Assessment".
Ph.D. thesis. University of Bristol, Dept. of Civil Engineering.
and second hand information from
Zadeh, L.A. (1965). "Fuzzy Sets". Information and Control. 8, pp. 338-353.
>From my reading, I understand that "fuzzy sets" are said to be
generalization of original sets, called "crisp sets". In fuzzy set theory,
each point in the "Universe of Discourse" (which in crisp set theory is the
sample space) is assigned a number between 0 and 1 that defines the degree
of membership of a considered fuzzy set, e.g. "Tall". To me this appears
equivalent to define the set "Tall" using a conditional distribution?
Mathematically, I understand that one of the main differences between crisp
sets and fuzzy sets lies in the definitions of unions and intersections.
This, of course, may be a critical problem. Intersection between two sets
A, B is defined in terms of a new member function
m_AB = min(m_A, m_B);
where m_A and m_B are the member functions for the sets A and B,
respectively. Similarly, unions are defined in terms of a max-member
function. The (fuzzy) union and intersection is then seen to be based on an
independence assumption. Obviously, this has been discussed elsewhere, and
relaxed choices of appropriate operators to model unions and intersections
of fuzzy sets have been introduced (even such that "crisp" results are
obtained!). If the set A was fuzzy "Tall" and B was fuzzy "Small", an
independence assumption may seem strange. In some universes, I guess, one
might be perfectly happy with a "universe of discourse" in which being "not
Tall" definitely says nothing of the likelihood of being "Small", or vice
versa.
I have seen no requirements of the fuzzy definition of "belonging to sets"
that say that a point cannot belong to several sets (mutually exclusive
sets). Therefore if YOU in fuzzy logic refer to a given person, YOU can
state that he/she belongs to set "Tall" with certainty 0.82 and set "Small"
with certainty 0.64, which certainly is fuzzy. I have neither seen
requirements that say that the sum of (some) membership for a given point
should sum to one (i.e. should be collectively exhaustive). Therefore, the
probability of being "Tall" or "Small" might sum to something different from
one. In my Bayesian world, "Small" and "Tall" would therefore be
represented by two (conditionally) independent nodes --- with a common
parent.
I cannot see what cases fuzzy set theory is capable of modeling better than
Bayesian Network models. BN may readily model all kinds of informal
arguments --- even treated in a rich interpretation. Since many people
perform probabilistic assessments using fuzzy sets and advocate for the use
of these, I would appreciate some enlightening. What important aspects did
I overlook?
Thank you,
Peter Friis Hansen
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