Re: Fuzzy logic compared to probability

Fred A Watkins (fwatkins@hyperlogic.com)
Fri, 8 Mar 1996 22:31:19 +0100


sthomas@decan.com (S. F. Thomas) wrote:

>Darren J Wilkinson (D.J.Wilkinson@durham.ac.uk) wrote:
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>: S. F. Thomas (sthomas@decan.com) wrote:

>> [ ... MUCH deleted material ... ]

>: For a Bayesian, the
>: link is provided by consideration of symmetry and invariance, the
>: strongest form of which is given a name: "Exchangeability". This is the
>: notion that beliefs over a collection remain invariant under an
>: arbitrary permutation of elements of the collection. Such a notion leads
>: to a representation for the collection which separates belief into
>: belief about the "underlying true value", and "individual vatiation".
>: Weaker forms of symmetry often lead to similar representations. It is
>: these representations which allow understanding of underlying processes,
>: and allow future prediction.

>This reminds me of the software programmer who attempts to pass off
>a bug as a feature. My objection to the Bayesian schema is logically
>prior to the Bayesian paradigm itself. It is no defense of the
>Bayesian paradigm to argue from within the paradigm. It is
>an impregnable position, yes, but in the same way that defenders
>of solipsism are immune from attack. Subjectivism is a mighty
>fortress indeed, and remains the essential stratagem of de Finetti,
>Savage and the other neo-Bayesian authors of the 20th century. I
>certainly would not attempt to defeat Bayesianism on such
>ground, and I readily concede that as a work of axiomatic mathematics,
>the neo-Bayesian edifice is both impressive, and internally
>consistent. Where I have difficulty is making that edifice
>correspond to reality as I apprehend it. Subjective prior belief,
>no matter how sophisticated the mathematics, or how clever the
>trotting out of such concepts of "exchangeability", remains an
>artful dodge where the central problem of inference is concerned,
>which is, how to characterize what *the data* say about some unknown
>probability distribution of interest. Throwing in a statistician's --
>or a "user's" -- prior belief into the mix, is still, in my opinion,
>sidestepping the real question. And arguing for the *necessity*
>of so doing, not to mention the *goodness* of so doing, is the
>Bayesian equivalent of turning a bug into a feature.

Chuckle. Nice shot.

>: There are two fundamental misconceptions at the heart of fuzzy theory.
>: The first is that uncertainty can be adequately understood without the
>: notion of probability.

>Maybe, or maybe not. I would leave it to the proponent to show.
>Reality stays the same. Paradigms come and go.

Hearty guffaw. Quite right, professor.

>: Probability theory exists precisely as a language
>: for the understanding of uncertainty.

>Uncertainty of a certain kind, however.

Har! Three times running!

>I happen to agree that the attempt to assert fuzziness as something
>totally unrelated to probability is ultimately misguided. But
>that is not the same thing as saying that fuzzy *is* probability.
>As I have argued, here and elsewhere, there is rather a sort of
>duality linking fuzzy and probability, in exactly the same
>way that likelihood and probability are distinct, but related
>concepts. In any case, fuzzy and probability do not necessarily
>compete, except when both retreat into a sort of solipsistic
>subjectivism, where everything becomes a matter of personal taste.

Well, here I might differ somewhat. I say the difference is
definition, or lack of it. Perhaps the numerics are similar (the big
demand the probabilists must have and the fuzzyists can reject is
summability, which has many implications) but the concepts differ.

To baldly assert that probability correctly models ill-defined events
(ultimately every event in reality, if we are strict Boolean
logicians) takes an act of faith. But the Bayesians are good at
belief, so I am told.

> [ ... MUCH MORE deleted data ... ]

>The essential stratagem revealed -- making of a bug, a feature.
>I do not deny the importance of subjective belief in certain
>situations. But to take this ounce of truth, and to make of it
>the whole inferential meal, is, again, sidestepping the essential
>problem of inference, which is to characterize what *the data*
>say.

Yes!

>: --
>: Darren Wilkinson - E-mail: d.j.wilkinson@durham.ac.uk
>: <a href=http://fourier.dur.ac.uk:8000/djw.html>WWW page</a>

>Regards,
>S. F. Thomas

Congratulations. It's good to see that someone can see through the fog
a bit.

Fred A Watkins