# Re: does fuzzy bound probability?

Subject: Re: does fuzzy bound probability?
Date: Thu Nov 23 2000 - 17:22:21 MET

Dear Scott,

It looks like when you talk about a "fuzzy number", what you really have in
mind is an interval of possible values of a certain quantity and a number in it
(like "the most probable value" in this interval.)

Of course, if you know an interval for X and an interval for Y, then the sum
belongs to the interval X+Y irrespective of the probability distribution that
we have on both intervals.

This is a known fact: that interval computations provide guaranteed estimate.

> Date: Tue, 21 Nov 2000 09:57:23 -0500
> From: Scott Ferson <scott@ramas.com>
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> Subject: Re: does fuzzy bound probability?
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>
> Actually, fuzzy *arithmetic* (as distinct from fuzzy logic) does NOT
> assume positive association between the operands. We're talking
> here about (say) adding together or subtracting fuzzy numbers. As
> an example, suppose
>
> X = [3, 6, 8]
> Y = [1, 2, 3]
>
> where [a,b,c] is a fuzzy number with peak b and base from a to c.
> Then X + Y is [4, 8, 11], and X - Y is [0, 4, 7]. Neither answer
> makes any assumption about association or correlation of X and Y,
> as they are simply level-wise generalizations of interval analysis.
> This is (part of) the reason that analysts have thought that Kaufmann
> and Gupta's fuzzy arithmetic might produce bounds on the answer
> obtained in a probabilistic analysis.
>
> Scott Ferson
> Applied Biomathematics
>
>
>
> WSiler@aol.com wrote:
>
> > In a message dated 11/21/00 5:10:46 AM Central Standard Time,
scott@ramas.com
> > writes:
> >
> > << Because (standard) fuzzy arithmetic makes no assumptions about the
> > > dependence or independence among quantities, it has been suggested
> > > that fuzzy arithmetic might be able to provide bounds on probability
> > > distributions in cases where the dependence among input variables
> > > cannot be specified empirically.
> > >>
> >
> > I'm not sure what basis was offered for the statement that fuzzy logic
makes
> > no assumptions about independence, but I'm afraid that the statement is not
> > true. Standard Zadehian min-max fuzzy logic assumes implicitly that the
> > operands of the logical operations AND and OR are positively associated as
> > much as possible, just as the probabilistic AND (a*b) and OR (a + b - a*b)
> > assumes independence (zero association) and the bounded sum and difference
> > logic AND (max(0, a + b - 1)) and OR (min(a +b - 1)) assume maximum
negative
> > association. Jim Buckley and I have a paper or two in FS&S which discuss
this
> > point and present a family of multivalued logic parameterized in terms of
the
> > correlation coefficient between the operands, based either on neccessity (a
> > AND NOT a) or past history.
> >
> > Consequently, any manipulations which employ fuzzy logical operations (such
> > as the extension principle) implicitly make an assumption about the
> > independence of the operands. That would seem to include fuzzy arithmetic.
> >
> > William Siler

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