Dear Prof. Zadeh (and others),
My view is that the notion of imprecise probabilities is not well-defined.
It seems that you are imagining something like, before the experiment is
performed, the probability distribution is drawn randomly from some set of
distributions, then the experiment is performed based on that distribution.
But it seems to me that these intermediary "probabilities" aren't really
probabilities then, they are model parameters. There will still be a final
distribution over the values of the outcome of the experiment, which in your
model is found by marginalizing over the "random probabilities".
To me, the essence of probability is that some event occurs according to a
fixed distribution. A probability distribution can't apply to only one
experiment. Whatever event that is being observed either will or will not
occur with some stationary distribution. If it does have a distribution,
then it is impossible to distinguish any particular generating sequence of
random variables. Any model that yields that final distribution when
marginalized is possible. You might be able to talk about the distribution
evolving in time.
Regarding your probability questions (c) and (d), I don't think you can
specify the marginal distributions of the "probabilities," aij, in advance.
There might not be a joint distribution that satisfies those marginals, e.g.
if X takes values in {1,2,3}, the probabilities p1, p2, and p3 can't all be
in an interval above 0.5 for example.
It would be possible to find the joint distribution that results from
restricting the probabilities to certain (admissible) intervals. In the
uniform case, the joint density would occupy the intersection of a hypercube
with the simplex. Expectations would be calculated by integrating over this
polytope, normalizing by its volume. A weighted integration could be
performed in the triangular case. I think intersection of linear varieties
with hypercubes are called zonotopes, and there may be formulae for their
volumes.
Kind regards,
Jason
-----Original Message-----
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]
Behalf Of Lotfi Zadeh
Sent: Wednesday, November 09, 2005 5:15 PM
To: uai@ENGR.ORST.EDU
Cc: [EMAIL PROTECTED]
Subject: [UAI] Imprecise Probabilities--A simple and yet computationally
nontrivial problem
Most real-world probabilities are imprecise. For this reason, as we
move further into the age of machine intelligence and mechanized
decision-making, the problem of how to deal with imprecise probabilities is
certain to grow in visibility and importance.
A major contribution to the theory of imprecise probabilities was
Peter Walley's 1991 book "Statistical Reasoning with Imprecise
Probabilities," London: Chapman and Hall. Since then, considerable progress
has been made. And yet, what is obvious is that the problem of computation
with imprecise probabilities is intrinsically complex and far from
definitive solution.
As a case in point, I posted to the UAI list (September 22, 2005) a
seemingly simple problem which does not have a simple solution. In the
following, a broadened version of the problem is presented. The simplest
version is Problem (a). My perception is that even this simple problem is
computationally nontrivial. Do you have a simple solution? Do you have any
solutions to Problems (b), (c) and (d)?
Problem:
X and Y are random variables taking values in the set (1, 2,
...,n). The entries in the joint probability matrix, P, are of the form
"approximately aij," where the aij take values in the unit interval and add
up to unity. What is the marginal probability distribution of X? Four
special cases: (a) "approximately aij," is interpreted as an interval
centering on aij; (b) "approximately aij," is interpreted as a triangular
fuzzy number centering on aij; (c) "approximately aij," is interpreted as a
uniform probability distribution over an interval centering on aij; and (d)
"approximatelt aij," is interpreted as a triangular probability density
function centering on aij.
With warm regards to all
Lotfi
--
Lotfi A. Zadeh
Professor in the Graduate School, Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720 -1776
Director, Berkeley Initiative in Soft Computing (BISC)
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