Re: Possibility theory and Bayesian inferences... some questions

On 25 Jul 2006 03:49:54 -0700, M. Sifniotis wrote:

I am trying to understand the difference between Bayesian thinking and
possibility theory when evaluating an agent's uncertainty. I would
like to use possibility theory to describe how certain (or not) an
expert is of a statement A. This statement A is not based on complete
knowledge, in fact the full evidence surrounding A can never be
accumulated. From what I've read I believe possibility theory would be
the way to go.

However from what I have understood Bayesian requires a lot of
experimental results?

I've started a conversation with a Bayesian mathematician friend who
insists possibility theory is junk. He provides the following example:
-------------
I have one hundred people in a room, half with blue eyes and half with
brown, One is selected by an unknown process and is about to come
through the door.
Let P be the statement that the person about to become visible has a
Blue left eye,
Let Q be the statement that the person about to become visible has a
Blue right eye,
Let R be the statement that the person about to become visible has a
Brown left eye,
Let S be the statement that the person about to become visible has a
Brown right eye,

The following statements are semantically correct:

1. The believability of P is about ½. It might reasonably mean that
we are about as committed to P as to its negation.
2. The believability of Q is about ½.. For the same reasons.
3. The believability of R is about ½.. For the same reasons.
4. The believability of S is about ½.. For the same reasons.
5. The believability of P OR Q is also about ½. This is because if P
is true then almost certainly Q is true too.
6. The believability of P OR R is pretty close to 1. That is, the
proposition that either his left eye is blue or it is brown, looks
pretty certain to be true.

Given the hypothesis about the hundred people we can pretty much say
that his left eye is bound to be either blue or brown. By the above,
using possibility:
Poss(P OR Q ) = max{Poss(P),Poss(Q) }. ½ for Poss(P OR Q ) , Poss(P
OR R ) but also for Poss(P OR R ) which is inconsistent with assumption
6!.

In the above you are mixing different things. Possibility is an uncertainty
measure, as well as probability is. When you are talking about P, Q, R, S
(events) you have to specify what you mean: Pr(P) or pos(P)? They are not
equal. Even numerically, they are not. In your case pos(P)=pos(Q)...=1,
because it is quite obvious that a person can *possibly* have whatever
color of eyes.

---------------
He then goes on and shows how P(P OR R)=1 for Bayesian thus making it a
logical choice.

I am trying very hard to understand about possibility theory and I
think I've grasped just about the basics. Could anyone from you give
me a _simple_ example with _simple_ inferences for Poss and Nec and how
these would compare to a Bayesian reasoning? Preferably by using
everyday language statements. For example, the one above, does it have
a correct solution? Dubois and Prade papers sometimes appear too
complicated and I have not found yet a simple and understandable
introduction to possibility theory. I guess there may be some in the
book by Dubois, Prade but the library has me on a waiting list for
it...

The essence of Bayesian approach is inversion of conditional probabilities
Pr(B|A)---inference--->Pr(A|B).

The Bayes' theorem

Pr(A|B)=Pr(B|A)Pr(A) / Pr(B)

has an equivalent formulation in possibility theory:

pos(A|B) = pos(B|A)
nec(A|B) = nec(~B|~A)

So the principles of Bayesian inference (a posteriori knowledge) can be
applied for systems dealing possibility. But you should be careful with the
applicability of. For the example you gave, possibility theory is too
rough. You won't get anything better than "it is fully possible", because
the experiment you constructed is clearly based on *random* choice of a
person from some set. It is a classical probabilistic model, where
probability theory works best. To make fuzzy useful you need a *fuzzy*
choice. For example, consider somebody trying to determine someone's else
eye color from a distance. You ask him, is it brown? And he answers "I am
unsure, but it looks so." That would be a possibility measure:

pos (color=brown | person=x) = 0.8
pos (color=blue | person=x) = 0.2

Now

pos (color=brown OR color=blue | person=x) = 0.8
pos (color=brown AND color=blue | person=x) <= 0.2

(note only an inequality).

pos (color/=brown | person=x) = 1 - nec (color=brown | person=x)

(it is as possible, as the opposite isn't necessary)

------------------------
There is a short summary of results of the possibility theory on my page:

http://www.dmitry-kazakov.de/ada/fuzzy.htm

A possibilistic formulation of Bayesian classifiers and some rules of
inference based on that you can find here:

http://www.dmitry-kazakov.de/fuzzy_ai/on_fuzzy_machine_learning.htm

--
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de
.

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