>The "generalizations" you always talk of hinge around the concept of
>t-norm and t-conorm. But: as far as I know, the choice of these norms
>is _completely arbitrary_, in contrast to the Bayesian framework,
>where all operations are founded in basic "axioms".
What I like about probability is not that it is axiomatic per se, but
that the axioms give me a semantics for relating the logic to
outcomes in the world. After all, there are mathematical axioms for
t-norms, too. However, no one to my knowledge has put forward a more
satisfying theory than decision theory for how to evaluate whether a
given t-norm model is an adequate model of a phenomenon.
On the other hand, if (I acknowledge that can be a big "if"!) I have
a proposition satisfying the clarity test, to which I assign
probability p, then I expect in approximately 100p% of the cases in
circumstances covered by the model that the proposition will be true.
More than that, I can quantify the likelihood, for different numbers
of trials, of different amounts of deviation from 100p%. This gives
me a means of evaluating scientific hypotheses about the likelihood
of different candidate probability models for a phenomenon, given a
set of observations. See Howsen and Urbach, Scientific Reasoning: The
Bayesian Approach. This is a satisfying semantics because it gives
me a pragmatically useful policy for validating scientific theories.
> > Important contributions to the theory of such nets were made by Dubois and
>> Prade. For recent results see Benferhat, Dubois, Kaci and Prade (Proc. Of
>> UAI'99, 57-64, Morgan Kaufmann, 1999.)
>
>I've worked with people in this community, and they are masters in
>"tuning" the above-mentioned arbitrariness to get the results they
>needed. (And they know it...)
If you give me a study in which you demonstrate convincingly, based
on empirical evidence, that a fuzzy system out-performs a Bayesian
system on a given class of problems when computational tractability,
accuracy, and usability are taken into consideration, then I can give
you a good solid decision theoretic argument for why you should
install the fuzzy and not the Bayesian system. As an engineer,
that's exactly the advice I would give you.
A student of mine did exactly this several years ago for a class
project. He compared fuzzy rules with a Bayesian linearized
controller for ship autopilot. The fuzzy rules did better than the
linearized controller, demonstrating that a semi-parametric
approximation to the right problem can beat an optimized solution to
the wrong problem.
But notice that my semantics in the above argument is decision
theoretic. I am reasoning at the problem-class level. I am
hypothesizing that the utility, averaged across exchangeable problem
instances, of the fuzzy system output is higher than the utility,
averaged across exchangeable problem instances, of the Bayesian
system output. If that is the case, then go with fuzzy.
As a scientist, I still don't understand what a t-norm means in the
world, except that it is a semi-parametric model that a good engineer
can "tweak" to approximate results that have high expected utility
for a given class of problem. As a scientist, I would want to
understand what it is about our hypothetical class of problem that
makes this kind of semi-parametric model such a good approximation.
That is, I would want to identify a scientifically validated decision
theoretic model that it was approximating. If I could do that, and
prove some theorems about the conditions under which the
approximation was tractable and accurate, then I'd feel much better
about advising my client to buy the fuzzy system. A solid
theoretical justification is much better in general than an empirical
regularity I don't fully understand.
Kathy
