GRAPHICAL MODELS

A popular graphical representation of probabilistic models in AI is the Bayesian network formalism, where a directed acyclic graph is used to specify a joint distribution over a set of variables $\tilde{X}$ [18]. Each node of a Bayesian network is associated with a variable X_i; the parents of X_i are denoted by $\mbox{pa}(X_i)$. This paper deals with variables X_i with a finite set of values.

Each variable in a Bayesian network is associated with a conditional distribution $p(X_i\vert\mbox{pa}(X_i))$. Such a graphical structure defines a unique joint probability distribution through the following expression [22]:

$$p(\tilde{X}) = \prod_i p(X_i \vert \mbox{pa}(X_i)).$$ (1)

Inferences with Bayesian networks usually involve the calculation of the posterior marginal for a queried variable X_q given evidence E [18].

Bayesian networks represent many independence relations among the variables in the network. These relations can be analyzed through the concept of d-separation: if $\tilde{Y}$ d-separates X from $\tilde{Z}$, then X and $\tilde{Z}$ are independent given events defined by $\tilde{Y}$ [22, page 117].

One difficulty with Bayesian networks is the requirement that all probability distributions must be precisely specified. Several non-probabilistic attempts have been made to relax the requirements of Bayesian networks through alternative theories of inference [12,25,26], or through interval-valued probabilities [2,11,16,15]. Interval representations have two problems. First, it is not always possible to apply Bayes rule to an interval-valued distribution and obtain an interval-valued posterior distribution [7,15]. Second, there is no unique, accepted way to define independence for interval-valued distributions [6]. Closed convex sets of distributions are also models for imprecision in probability values [1,15,24,28]. Closed convex sets of distributions have several advantages when compared to interval-valued probability because conditionalization and independence can be defined without technical difficulties. In this paper, closed convex sets of distributions are employed as fundamental entities that reflect perturbations and imprecision about stochastic phenomena.

One axiomatization of closed convex sets of distributions that is particularly concise and powerful is the Quasi-Bayesian theory of Giron and Rios [14]. This theory is summarized in the next section; several recent definitions and results, not present in the original theory by Giron and Rios, are incorporated in the presentation.