A Bayesian network represents a joint distribution through a collection of locally defined probability distributions. Consider a set x of discrete variables with a finite number of values. A Bayesian network defines a probability distribution through the expression [Pearl1988]:

p(x) = prod_{i} p(x_{i} | _{i})),

For any function f(), we use the abbreviation f_{i} for
f(x_{i}|_{i})). In this notation,
expression (1) can be rewritten as
p(x) = prod_{i} p_{i}. Evidence e refers to a set of values
for some variables in the network.

From a robustness perspective, it is convenient to understand sets of probability distributions as neighborhoods for base probability distributions. Call Gamma(p) the neighborhood of distribution p() [Wasserman & Kadane1992]; Gamma(p) is a convex set of distributions induced by p().

The local structure of a Bayesian network has been explored in several attempts to encode probability intervals into a joint distribution [Breese & Fertig1991, Cano, Delgado, & Moral1993, Chrisman1996b, Shenoy & Shafer1990, Tessem1992]. A general solution for inferences with local neighborhoods in a Bayesian network has been given in a companion technical report, for most of the known neighborhoods used in robust Statistics [Cozman1996]. Here we depart from this local model and seek global neighborhoods for Bayesian networks, which are appropriate to model global perturbations such as the effect of invalid independence assumptions. We concentrate on neighborhoods Gamma(p()) for the joint distribution.

In the next sections expressions for upper and lower expectations are developed. Consider a function u(x); we assume that u(x) = u(x) where x is a subset of the variables in x. The expressions indicate how to obtain upper and lower expectations for u() of this form.

© Fabio Cozman[Send Mail?]

Thu Jan 23 15:54:13 EST 1997