Exercise 10.7 - Removing leaves in BN20 networks

For question (a), note that given $\mathbf{\text{𝐳}}$, all components from $\mathbf{\text{𝐱}}$ are mutually independent, hence:

according to Figure 10.16.(b), while that computed w.r.t. Figure 10.16.(a) is:

This completes the proof.

For question (b), consider the following decomposition of the joint probability:

where ${𝜃}_{Z,Y}$ is defined from the correlation between disease $Z$ and symptom $Y$. If $Z$ and $Y$ are independent then ${𝜃}_{Z,Y}=1$.

Now:

By changing the prior over $Z$ from $p(Z)$ into $p(Z)\left[{\prod <!--\; FUNCTION\; APPLICATION\; -->}_{Y\in {\mathbf{\text{𝐗}}}_{\text{off}}}{𝜃}_{Z,Y}\right]$ we complete the proof.