Re: Fuzzy logic compared to probability

Josh Kuperman (ߠ\ݰjosh@racing.saratoga.ny.us)
Fri, 1 Mar 1996 21:21:18 +0100


First, let me state that I am a leyman and only doing this for the money.
No, uh, because I like the sound of my own voice.
No, uh, because I fell like disagreeing with someone.
No, uh, well, whatever.

In <4gh66g$r8@elna.ethz.ch> poncet@isi.ee.ethz.ch (Andreas Poncet) writes:

>In article <312B60FB.41C67EA6@colorado.edu>, Robert Dodier <dodier@colorado.edu> writes:

>what if we consider the Bayesian subjective view of probability, namely
>as a degree of belief in (or a degree of relevance of) a given
>proposition? Then it seems to me that a "membershipness" is in fact
>a CONDITIONAL PROBABILITY.

>As an example: instead of saying, e.g., about a specific man, that
>his height x has "tall"-membership, say, 0.8,
>one could express this equivalently as P(A|x) = 0.8,
>with the proposition A defined as
> A = "the man is tall"

Ah, but there is a distinction we must make here. Probabilities say if we
pick a man at random, or in the Bayesian case, take one particular man and
let someone look at him, then there is a such-and-such probability that he
is (considered) tall. What the fuzzy logic / fuzzy set theory states is
that a man of such-and-such a height belongs to the set of tall people to
such-and-such a degree. Any man of such length always belongs to that set
with the same degree of membership. I.e. he is not "pretty probably tall",
he "is pretty tall". Classical probability is based on frequency, Bayesian
on subjectivity. Fuzzy sets and fuzzy logic, like crisp sets and logic,
state objective truths. Fuzzy allows for these truths to be spread over a
wider spectrum than "NO"/"YES", but they are still objective and absolute.
To clarify: _any_ man of 180 cm is _always_ said to be "0.8 tall", rather
than "80% of the 180 cm men are tall". Instead of putting 80% of the men
inside the tall set and 20% outside, you put all of them at varying distances
from the center of the set.

>Fuzzy logic couldn't convinced me yet, for there seems to be much "ad
>hockery", as in the choice of the "tent" form, e.g., of membership
>functions. Why such a form? How can it incorporate prior information?
>(Probability theory decomposes the problem of specifying P(A|x) in two
>steps: the prior P(A) and the likelihood p(x|A).)

A membership function does not need to have a specific form, does it?
Take a function, any function. This function can be considered a fuzzy
membership function if you wish. It can be, I don't know, y=x^2, say.
Or a gaussian. Since, in applications, many fuzzy sets are based on
statistical analyses of populations or similar, they often get a gaussian
curve, but they don't need to. The problem lies in selecting your function
so that it, as you say, incorporates the information you want to describe.
While this may not be a simple problem, it is certainly solvable.
What you need to do once you've found a function is define set operations
in terms of this function, union, intersection, complement, and so on.
You need, in specific, to find a pair of functions, one of which can be
used for intersection, one for union. These functions must meet certain
requirements, they must be a t-norm/t-conorm pair. One such pair is
Max (for intersection) and Min (for union), there are others more complex.

>Furthermore, how to consistently specify JOINT membership functions of DEPENDENT
>characteristics (such as "tall"-"heavy")? Again, using probability
>theory guarantees consistent answers.

IF he is tall, THEN he is heavy, you mean?
Well, looking at it logically, that's T => H, right?
And since T => H <=> ~T OR H, all we then have to do is apply our
set complement and union, and hey presto! we've got a fuzzy set describing
the required statement.

>> It would appear that if the fuzzy logic concept of `degree of
>> truth' is the same as the Bayesian `degree of belief,' then
>> either fuzzy logic is the same as probability or else less
>> powerful (as it would have to be inconsistent).

I'd say fuzzy logic is no closer to probability than is crisp logic.
Logic in any form speaks of truths, probability of likelyhoods.

Disclaimer: I may be talking complete rubbish. I should be marking
assignments, but it's a lazy Saturday. Anyway, I'm not a
professor or anything, so I may be out of my league.

MVH,

/Uffe

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