Re: Transfoming probability distributions into fuzzy sets - can anyone help?

J. Lawry (enjl@PROBLEM_WITH_YOUR_MAIL_GATEWAY_FILE)
Mon, 17 Aug 1998 20:47:03 +0200 (MET DST)

WSiler (wsiler@aol.com) wrote:
: >Just a general comment that I think it is best to keep fuzzy logic and
: >probability theory as far removed as possible!
:
: I realize than many or most fuzzy persons share your viewpoint here, but I do
: not. Both probability distributions and membership functions look from
: different vantage points ar pretty much the same thing - uncertain numbers,
: whatever the cause of the uncertainty may be. A very few persons have worked
: seriously on the relationship between the two, which I think is unfortunate.
:
: There is, in any event, a quick (if incomplete and inaccurate) answer to the
: question. Probability density functions have area one; fuzzy numbers and
: membership functions usually have max value one. Simply normalize the
: probability distribution to a max value of one, and there is your fuzzy number
: or membership function. If the probability distribution is discrete, do the
: same thing, and there is your discrete fuzzy set.
:
: The question of the circumstances under which that simple answer is wrong is
: interesting, and comments on this would be welcome. Am I wide open?
:
: William Siler

While I agree that we should not rule out a relationship between fuzzy
sets and probability ( indeed I am a strong advocate of probabilistic
semantics for fuzzy sets) I do not agree that we should take probability
distributions of random variables (normalised or not) as membership
functions of fuzzy sets. The former quantify uncertainty regarding the
value of a random variable and the other vagueness of definition.

Instead it seems to me the most coherent probabilistic model of fuzzy sets
is that of random sets. Put simplistically a random set is a set valued
variable associated with which is a probability distribution (a mass
assignment). If we then interpret vagueness as uncertainty regarding a crisp
definition (a controversial view but one I believe is tenable) random sets
provide an ideal formalism. Now for a random set S into the power set of
some (say finite) universe U the one point coverage of S is given by Pr(x
in S) for x in U. This can be taken as the membership function of an
associated fuzzy set.

Changing track slightly and returning to the original question about
generating fuzzy sets from probability distributions I think that the
answer lies in the probability of fuzzy events. This approach
proposed by Prof Jim Baldwin assumes that the inferred fuzzy set should be
descriptive of the probability distribution in the sense that if we condition
of the fuzzy set (relative to a fixed prior) we will obtain the
probability distribution. In otherwords, if Q is the probability
distribution then we require to find a fuzzy set f such that

For x in U Pr(x|f)=Q(x)

(where Pr(x|f) is relative to the uniform prior if there is no prior
information)

This is an extension of the intuitive idea that ,for example, given a
standard dice problem (i.e.a fair dice) the set even_number={2 4 6} is
descriptive of the distribution Pr(2)=1/3,Pr(4)=1/3,Pr(6)=1/3.

Assuming the mass assignment theory of the probability of fuzzy events (a
theory consistent with the random set interpretation I outlined above)
then this method requires that the probability distribution Q should
correspond to the least prejudiced distribution of the fuzzy set f. This
constraint can be satisfied uniquely for any finite universe. If anyone is
interested in the details I can provide some references.

Regards

Jonathan Lawry

-- 
Dr Jonathan Lawry,
AI Group,
Dept. Engineering Mathematics,
University of Bristol,
Queens Building,
University Walk,
Bristol, BS8 1TR, UK

Email:j.lawry@bristol.ac.uk Tel:+44 117 928 8184

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