Consider the causal network of Figure 11.12 (page 513). For each part, explain why the independence holds or doesn’t hold, using the definition of d-separation. The independence asked needs to hold for all probability distributions (which is what d-separation tells us).

(a) Is J independent of A given {} (i.e., given no observations)?

(b) Is J independent of A given {G} (i.e., given only G is observed)?

(c) Is J independent of A given {F}?

(d) Is J independent of A given {G, F} (i.e., given only G and F are observed)?

(e) Is G independent of J given {}?

(f) Is G independent of J given {I}?

(g) Is G independent of J given {B}?
(h) Is G independent of J given {B, I}?
(i) What needs to be observed, and what needs to be not observed for G to be independent of J? Give a complete characterization.