I apologize for the length of this post, but some of the
issues deserve more careful consideration.
Herman Bruyninckx wrote:
> Bayesian theory is fully consistent: there is only _one_ way to do
> your algebra.
It is perhaps a bit misleading to say that because Bayesian theory
is consistent that there's only one way to do a Bayesian analysis
for a problem. Come on! In fact, if the use of subjective priors
is involved, then there are as many different analyses as there
are analysts.
> While fuzzy logic has an infinite amount of possible
> computational rules: all hinges around the fact that there are
> infinitely many ways to define t-norms or t-conorms, and there is no
> ``first principle'' that tells you which one to choose. Therefore, you
> can write an infinite number of papers on the same real system and the
> same data :-) (Which is what many people do indeed :-( )
In principle, the different t-(co)norms correspond to different
underlying mechanisms that govern how the inputs should
be combined. This should rightly be considered flexibility,
rather than merely loose definition.
> Hence, fuzzy logic is indeed more general; in fact it is too general to
> be still called a scientific paradigm (because of the above-mentioned
> indefiniteness of its calculus).
It does seem clear that a fuzzy approach is more general because
it is axiomatically weaker, but I don't think it's quite fair to conclude
it's uselessly general. Perhaps a more generous interpretation is
that it has *wider applicability* than a strict probabilistic approach.
Concomitantly, we'd expect it to be a less powerful theory in
situations where the axioms of probability theory are satisfied.
Peter Friis Hansen wrote:
> I have been asked to write a one-page summary on fuzzy sets --- yet another
> area of which I have no knowledge --- and I therefore would be interested in
> some clarification on what cases fuzzy set theory is capable of modeling
> better than Bayesian Network models.
A brief comparison of fuzzy arithmetic versus a traditional
probabilistic approach using Monte Carlo methods appears
on http://www.ramas.com/riskcalc.htm#fuzzy Although it
focuses on *arithmetic* modeling rather than logic or set
theory, it includes a synoptic review of the reasons one
might chose one approach over the other.
> From my reading, I understand that "fuzzy sets" are said to be
> generalization of original sets, called "crisp sets". In fuzzy set theory,
> each point in the "Universe of Discourse" (which in crisp set theory is the
> sample space) is assigned a number between 0 and 1 that defines the degree
> of membership of a considered fuzzy set, e.g. "Tall". To me this appears
> equivalent to define the set "Tall" using a conditional distribution?
No, fuzzy membership is definitely not the same thing
as conditional probability.
> Mathematically, I understand that one of the main differences between crisp
> sets and fuzzy sets lies in the definitions of unions and intersections.
> This, of course, may be a critical problem. Intersection between two sets
> A, B is defined in terms of a new member function
> m_AB = min(m_A, m_B);
> where m_A and m_B are the member functions for the sets A and B,
> respectively. Similarly, unions are defined in terms of a max-member
> function. The (fuzzy) union and intersection is then seen to be based on an
> independence assumption.
On the contrary, using min and max respectively for
conjunction and disjunction corresponds to an
assumption of perfect *positive* dependence, and
*not* independence. As Bruyninckx's criticism
suggests, however, you are not restricted to such an
assumption. Modern fuzzy approaches allows one to
use a variety of t-(co)norms to represent different
kinds of dependence or interaction.
> I have seen no requirements of the fuzzy definition of "belonging to sets"
> that say that a point cannot belong to several sets (mutually exclusive
> sets). Therefore if YOU in fuzzy logic refer to a given person, YOU can
> state that he/she belongs to set "Tall" with certainty 0.82 and set "Small"
> with certainty 0.64, which certainly is fuzzy. I have neither seen
> requirements that say that the sum of (some) membership for a given point
> should sum to one (i.e. should be collectively exhaustive). Therefore, the
> probability of being "Tall" or "Small" might sum to something different from
> one.
First, remember that it's not the *probability* of being
tall or small. It's not a probability at all. It's something
else, sometimes called "possibility", which measures the
degree something is true (not its frequency, or even one's
belief that it's true).
Second, you should be careful about characterizing the
sets as "mutually exclusive" which is a phrase that recalls
Boolean logic. In a fuzzy set theory, the set of tall people
and the set of small people could well be not mutually
exclusive. I'm tall for a jockey, but pretty small for a
basketball player. It makes a difference what the sets
were constructed to represent.
On the other hand, one certainly can model mutually
exclusive sets in fuzzy logic (with membership functions
that have non-overlapping supports), and their behavior
will follow what you'd expect from ordinary Boolean logic.
After all, fuzzy logic generalizes ordinary logic, it doesn't
contradict it.
> I cannot see what cases fuzzy set theory is capable of modeling better than
> Bayesian Network models. BN may readily model all kinds of informal
> arguments --- even treated in a rich interpretation. Since many people
> perform probabilistic assessments using fuzzy sets and advocate for the use
> of these, I would appreciate some enlightening. What important aspects did
> I overlook?
Well, it is a real question whether there are many (or even
any) cases where probability cannot offer a reasonable model.
The theory is fairly mature and the tools and methods are
very rich. To some extent, the comparison should be about
which approach is more natural. Which provides a more *useful*
framework for considering the issues of interest.
I think what you may have overlooked is that not everything
fits neatly in the underlying Boolean logic that probability
depends on. Vagueness, for instance, is the classic example.
If our friend Mark is of intermediate height, then the truth of
a statement like "Mark is tall" is not really one, nor is it really
zero. There is no fact of the matter, only a degree to which
it is true. This situation, in which there are inescapably
borderline cases, constitutes what philosphers call vagueness.
As such, it cannot be handled properly by any Bayesian
approach because probability theory (including Bayesian
methods of course) generalize the Law of the Excluded Middle
that categorically states that either Mark is tall or he's not tall.
(Or, if it is handled properly, it isn't handled *conveniently*
by probability.) If, for some statements, you don't want to
use this assumption, then perhaps you'll decide it's better
not to use Bayesian methods or probability theory.
In risk analysis, the example might use the vague term
"poisonous" for a suite of substances that range in effect
from making you chronically sick to killing you in seconds.
Insisting that there always be bright-line definitions that
enable one to say definitely and unequivocally whether
a person is tall or a substance is poisonous does a real
violence to the language and forces all terms to take on
unnatural, technical meanings. What is potentially useful
about the fuzzy approach is that it permits an analyst to
make calculations in a way that avoids this violence yet
gets quantitative results that account for the underlying
vagueness. It can faithfully carry the vagueness along
through the calculations.
> In some universes, I guess, one might be perfectly happy with a
> "universe of discourse" in which being "not Tall" definitely says
> nothing of the likelihood of being "Small", or vice versa.
Again, it's not that it "says nothing" (that's another veiled
appeal to independence). It's that the sets are saying
different things altogether. For instance, I can be in
the set small to some degree and simultaneously in the
set tall to another (perhaps lesser) degree. So if we're
needing to make a calculation involving the count of
people in these sets, the answer might reasonably
depend on whether you're selecting sides in basketball,
picking jockeys, or designing automobile interiors.
The alternative available to non-fuzzy methods has been
to set a bright-line threshold that says who is a member
of the set and who isn't. But in situations where vague
terms like "poisonous" or "endangered" play an integral
role, this strategy may not be very useful. A species with
99 individuals is endangered and therefore deserves
special protection under the law, but a species with 101
is not endangered and therefore gets no protection.
Such a strategy might drive conservationists to kill a couple
just to make it endangered. Clearly, something has gone
awry as a result of not taking the vagueness into account.
Vagueness is just one of several domains where probability
has limited usefulness. This is also the case in reasoning
about fictional discourse, and perhaps for some kinds of
measurement error that violate the usual assumptions of
small error magitudes, randomness and independence.
Scott Ferson <scott@ramas.com>
Applied Biomathematics
631-751-4350, fax -3435
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