The issue of definition of a Bayes net has drawn a large number of
insightful comments. However, there are some important points that have not
been accorded adequate attention.
First, it should be recognized that a Bayesian net is an instance of a more
general structure which may be called a constraint net. In such nets, a
link between two nodes indicates that the node variables are related by a
joint or conditional constraint which is drawn from a family of generalized
constraints that includes probabilistic constraints as a special case. The
counterparts of product and sum in probabilistic computations are the
operations of t-norm and t-conorm. A class of constraint nets which in some
sense is dual to that of Bayes nets is the class of possibiistic nets.
Important contributions to the theory of such nets were made by Dubois and
Prade. For recent results see Benferhat, Dubois, Kaci and Prade (Proc. Of
UAI'99, 57-64, Morgan Kaufmann, 1999.)
A significant issue which is not met head on in the theory of
Bayes nets is related to the fact that, in most realistic settings,
probabilities and probability distributions are known approximately
rather than exactly. Computations with imprecise probabilities are, in
general, much more complex than with probabilities which are
exact. (In-depth results on computation with imprecise probabilities
may be found in Walley's treatise "Statistical Reasoning with
Imprecise Probabilities," (Chapman and Hall, 1991.) As an
illustration, assume that X and Y are normally distributed random
variables with interval-valued means, variances and covariance which
define the uncertainty of the joint distribution. If A and B are
specified events in R^2, computation of conditional probabilities and
verification of whether or not A and B are independent is
prohibitively complex.
Another significant issue relates to the brittleness of
probabilistic computations. Thus, if A, B and C are events, and
P(B|A)=1 and P(C|B)=1, then P(C|A)=1. But if P(B|A) and|or P(C|B) are
not exactly equal to unity, then, counterintuitively, no matter how
close P(B|A) and P(C|B) are to unity, all that can be said about
P(C|A) is that it is between zero and one. The same applies to P(C|A),
P(C|B), and P(C|A,B), under minor assumptions. These possibilities
have been noted in the literature of uncertainty management in expert
systems.
Underlying the cited issues is the brittleness of the concept of
independence. Slightest changes in values of probabilities can shift the
status of the relationship between A and B from independence to dependence.
Thus, if P(A) P(B) differs from P(A,B) by an epsilon, are A and B
independent? The brittleness of probabilistic computations draws attention
to a basic question: Are probabilistic computations in the context of
Bayesian nets brittle or robust? Is there cumulation of imprecision when
imprecise probabilities propagate through a Bayesian net? Answers to these
questions are complicated by the fact that bounds on imprecision are
determined by the worst case scenario. The problem is that the probability
of worst case scenario is hard to assess.
