LOCALLY DEFINED QUASI-BAYESIAN NETWORKS

This section defines Quasi-Bayesian networks that are generated from local models associated to a directed acyclic graph:

Definition 2   A locally defined Quasi-Bayesian network is a directed acyclic graph associated with: (1) either a single conditional distribution $p(X_i\vert\mbox{pa}(X_i))$ or a local credal set $K(X_i\vert\mbox{pa}(X_i))$ for each variable X_i, (2) a collection of irrelevance relations, and (3) a method for the combination of local credal sets.

A joint credal set that satisfies all constraints and relations in a Quasi-Bayesian network is called an extension of the network.

The rationale for this definition is as follows. In a standard Bayesian network, irrelevance and independence constraints are implicit in Expression (1); this expression guarantees that a variable is independent of all its non-descendants given its parents [22, page 119]. There is no analogue to Expression (1) in Quasi-Bayesian networks. Many extensions may satisfy all graphical d-separation relations in a network. It seems more appropriate to ask a decision maker to explicitly indicate which qualitative constraints are to be enforced in a Quasi-Bayesian network, and to ask for irrelevance constraints instead of independence constraints, because irrelevance and independence are not equivalent in Quasi-Bayesian models (Section 2.2).

The key fact is that a directed acyclic graph and a collection of local credal sets may admit more than one extension; the next sections investigate two important types of extension.