NATURAL EXTENSION

Type-1 extensions are not the only possible extension of a locally defined Quasi-Bayesian network. The natural extension of the network is the largest set of joint distributions compatible with local credal sets and irrelevance relations in the network. This terminology has been sugested by Walley [28, pages 453, 455], who explores properties of natural extensions but does not focus on multivariate structures.

A Quasi-Bayesian network is defined by quantitative constraints on probability values and by qualitative statements of irrelevance and independence. The quantitative constraints that define a credal set $K(X_i\vert\mbox{pa}(X_i))$ are denoted by $C_l \left[ p(X_i\vert\mbox{pa}(X_i)) \right]$.

The objective of this section is to investigate and exploit the representation of qualitative statements of irrelevance and independence in natural extensions, particulary statements that involve variables and their nondescendants. Many different natural extensions can be created for a given directed acyclic graph through different statements of irrelevance.

The algorithms focus on irrelevance and independence conditional on the nondescendants of a node. This strategy follows common practice in Bayesian networks (which are based on the agreement between d-separation and irrelevance/independence [22]); for natural extension, this strategy has a simple justification as follows. When stating irrelevance/independence relations among variables, it is important to guarantee that a natural extension can actually be constructed. Incompatible relations can lead to an empty natural extension. One strategy that always produces valid natural extensions is to rely on graphical d-separations as the source of irrelevance/independence relations, because there is always at least one standard Bayesian joint distribution that complies with all constraints. This rationale suggests that irrelevance/independence relations among variables and their nondescendants are of primary interest.