SPECIFYING CONDITIONAL CREDAL SETS SEPARATELY

The following algorithms assume that constraints on conditional distributions are defined separately for each value of the variable's parents. This means that, for any variable X_i, the constraints $C_l[p(X_i\vert[\mbox{pa}(X_i)]_{k_1})]$ do not interfere with the constraints for $C_l[p(X_i\vert[\mbox{pa}(X_i)]_{k_2})]$ when $k_1 \neq k_2$. This restriction makes sense both during elicitation of models and representation of constraints, and the following derivations exploit this restriction to generate inference algorithms.

Consider first the quantitative constraints $C_l \left[ p(X_i\vert[\mbox{pa}(X_i)]_k) \right]$. Because all local credal sets have a finite number of vertices, all constraints $C_l \left[ p(X_i\vert[\mbox{pa}(X_i)]_k) \right]$ are linear in $p(X_i\vert[\mbox{pa}(X_i)]_k)$. Because the value of $\mbox{pa}(X_i)$ is fixed in every constraint, all constraints are of the form:

$$\sum_{j=1}^{\vert\hat{X_i}\vert} \gamma_{ijkl} p(X_i = X_{ij... ...\mbox{pa}(X_i)]_k) \leq \gamma_{i0kl} p([\mbox{pa}(X_i)]_k),$$ (2)

where $\gamma_{ijkl}$ are constants that define the local credal sets. Note that these constraints are linear in $p(\tilde{X})$, because $p(X_i,\mbox{pa}(X_i))$ and $p(\mbox{pa}(X_i))$ are summations over $p(\tilde{X})$.

Note that, if a single distribution q is specified for variable Y_i, the only constraint imposed on the conditional distribution for Y_i is:

$$p(Y_i = Y_{ij}\vert[\mbox{pa}(Y_i)]_k) = q(Y_i = Y_{ij}\vert[\mbox{pa}(Y_i)]_k).$$