Re: Kosko Profile in IEEE Spectrum

Peter Hamer (pgh@bnr.co.uk)
Fri, 1 Mar 1996 23:36:29 +0100


In article <4gf8sc$4oh@newsbf02.news.aol.com> jbpir2@aol.com (JB PIR2) writes:
>Probabilist apologists who argue that Fuzzy Logic is the same as
>probability haven't got a clue as to what Fuzzy Logic is about. Kosko
>clearly proves that probability is a subset of Fuzzy theory. Even with his
>powerful arguments, there are still those who 'Just Don't Get It'.

The "logic" you use to manipulate your mathematical model is important, but
not the only important thing.

For example, Bayesian statistics and frequentist statistics use exactly the
same "logic" -- but the two schools have historically had some very bitter
disputes.

What fundamentally differentiates Bayesian and frequentist practice is
the real-world interpretation of the mathematical model. A consequence
of this is that frequentists are unable to model real-world situations
requiring any concept of belief.

If we completely ignore fuzzy *logic* for the moment. I find it difficult
to interpret membership functions as other than some form of statement of
belief. [Knee-jerk warning: I am not saying this is de facto the Bayesian
concept of belief.]

For example the membership function "is tall" captures the belief of its
creator in the "applicability" of the term to a person of various heights.

--- Checkpoint. How wrong am I so far? ---

A problem with this process is that it seems to be fundamentally a "tablets-
of-stone" approach. With the nature of the membership function appearing out
of nowhere [or as a result of some form of pure thought].

If you think about how you would [even in principle] try confirm the
"correctness" of a membership function it is hard to avoid the conclusion
that you require some method of experimental confirmation. [If you don't have
one, then its correctness is a metaphysical issue.]

Personally, I find it difficult to imagine any form of confirmation that
does not involve asking people what they think. Which rather naturally leads
to the value of the membership function at any point being the fraction of
those asked who agreed with the proposition (eg that a x-ft person "was tall").
[Or some monotonic transformation of this.]

Of course, in practice most membership functions will be somebodies guess-timate
of what people would have said if asked. But I think that the thought experiment
reveals the underlying semantics.

--- Enter slightly more controversial mode. ---

The conclusion that there is a legitimate Bayesian interpretation of membership
functions is hard for many of those familiar with Bayesian statistics to avoid.

.. unless suitably cogent arguments against are presented.

--- Fuzzy logic. ---

I have NOT discussed it. I'm not going to. Manipulating a mathematical model
using the rules of fuzzy logic will typically lead to different conclusions from
those reached by using the appropriate logic for probability.

The engineering choice of which logic is "more useful" to apply might rationally
be made by balancing: the evidence of performance on real-world problems; and the
theoretical desirability of properties such as consistency in a logic.

Personally I would like to see more comparisons of the performance achievable
by the two approaches. This may well require the different approaches to be
applied by those familiar with them. [I'm very bored by demonstrations that
method X (eg neural nets) outperforms linear regression -- on problems where
the only excuse for applying linear regression can be ignorance of non-linear
statistical methods.]

I would also like to see a discussion of why a particular fuzzy logic was chosen
out of the infinity of options. Pragmatics? Computational simplicity? Performance?

Peter