REPRESENTATION OF INDEPENDENCE RELATIONS

To guarantee that Y_i is independent of X_i given $\mbox{pa}(X_i)$, it is necessary to enforce that: (1) Y_i is irrelevant to X_i given $\mbox{pa}(X_i)$, and (2) X_i is irrelevant to Y_i given $\mbox{pa}(X_i)$. The first constraint has been addressed in the previous sections, but the second constraint introduces new complexities into the problem. For example, suppose that a variable X₅ has variables X₁, X₃, X₄ as nondescendants, and X₂ as parent. The second irrelevance condition requires that the credal sets K(X₁, X₃, X₄|X₂, X₅) and K(X₁, X₂, X₄|X₂) contain the same functions. The difficulty is that neither of these credal sets is directly specified on the network; there is no simple constraint that ties them together.