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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