

![0_image_0.png](0_image_0.png)

E given a parent context. For example:

$$\begin{array}{l}{{P(e|a,b,c,\overline{{{d}}},y,\overline{{{z}}})=P(e|a,b)}}\\ {{P(e|\overline{{{a}}},b,c,\overline{{{d}}},y,\overline{{{z}}})=P(e|\overline{{{a}}},c)}}\\ {{P(e|\overline{{{a}}},\overline{{{b}}},c,d,\overline{{{y}}},\overline{{{z}}})=P(e|\overline{{{a}}},c)}}\end{array}$$

In the belief network, the parents of E are A, B,C, D. To specify the conditional probability of E
given its parents, the traditional tabular representation (as in Figure 1) require 24 = 16 numbers instead of the 6 needed if we were to use the parent contexts above. Adding an extra variable as a parent to E doubles the size of the tabular representation, but if it is only relevant in a single context it may only increase the number of parent contexts by one.

We can often (but not always) represent contextual independence in terms of trees. The left side of Figure 3 gives a tree-based representation for the conditional probability of E given its parents. In this tree, internal nodes are labelled with parents of E in the belief network. The left child of a node corresponds to the variable labelling the node being true, and the right child to the variable being false. The leaves are labelled with the probability that E is true. For example P(e|a ∧ b) = 0.3, irrespectively of the value for C or D. In the tree-based representation the variable (E in this case)