Re: does fuzzy bound probability?

Subject: Re: does fuzzy bound probability?
From: Scott Ferson (scott@ramas.com)
Date: Wed Nov 22 2000 - 19:20:38 MET

Actually, fuzzy *arithmetic* (as distinct from fuzzy logic) does NOT
assume positive association between the operands. We're talking
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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