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

J. Lawry (enjl@PROBLEM_WITH_YOUR_MAIL_GATEWAY_FILE)
Mon, 17 Aug 1998 21:33:04 +0200 (MET DST)

WSiler (wsiler@aol.com) wrote:
: >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.
: >
: It is certainly true that "probability distributions quantify uncertainty
: regarding the value of a random variable", to say that "[membership functions
: of fuzzy sets characterize] vagueness of definition" is a quite unnecessary
: restriction on fuzzy sets. Having worked on real-world applications of fuzzy
: expert systems for some fifteen years now, I consider that fuzzy sets can
: characterize uncertainty of whatever origin, including both vagueness and
: values of random variables among many others.
:
: To assert that a normal distribution characterizes a numeric random variable
: subject to a large number of small errors amounts to a tautology, parameterized
: perhaps as a mean and variance. However, I can (and often do) characterize that
: same variable as a bell-shaped fuzzy number, paramaterized perhaps as central
: value and a hedge "roughly". There is no vagueness here, just an uncertainty as
: to precise value. In an expert system, "roughly 2" is a heck of a lot more
: useful than "2 +/- 25%".
:
: A list of the kinds of uncertainty which can be fruitfully represented by fuzzy
: quantities (e.g. truth values of scalars, fuzzy numbers, membership functions,
: truth values of rules, truth values of members of a discrete fuzzy set,...)
: would probably be quite long. If I'm not sure that a car is a Ford or a
: Chevrolet, that uncertainty is easily represented by the grades of membership
: in a discrete fuzzy set of car makes, for example.
:
: I'm not sure what latitude FRIL offers in the kinds of things which can be
: represented by fuzzy quantities, but I surely hope it covers more than vague
: definitions.
:
: William Siler
:

I would say that roughly_2 or about_2 are vague. Most often they model
uncertainty regarding the exact definition of a interval centred about 2.
This certainly would corresponds to vagueness according to the 'uncertainty of
definition' definition. On the other hand the restriction (X is roughly_2)
also conveys uncertainty regarding the exact value of the random variable
X. Indeed you have both types of uncertainty present here. This situation
is best modelled according a theory of the probability of fuzzy events.
More specifically, the contraint (X is roughly_2) induces a conditional
(posterior) probability distribution on X i.e Pr(X|X is roughly_2). This
distribution would take into account the prior distribution on X. Given
this distribution we can then make inferences regarding another fuzzy
event say (X is f) by determining Pr( X is f|X is roughly_2). This is
essentially how the semantic unification method in FRIL works.

Regarding your point about the usefullness of fuzzy intervals of precise
ones to say partition a universe, I completely agree that the former give a
degree of interpolation to your model absent when using the latter.
However, it seems to me that this comes directly from the uncertainty
regarding their exact definition i.e their vagueness

Regards

Jonathan

-- 
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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