d-separation [4+1* P]

a)

Prove or disprove:

1. $[1 P]$ $(X \perp Y,W\vert Z) ) \Rightarrow (X \perp Y \vert Z)$ .
2. $[1 P]$ $(X \perp Y \vert Z) \&  (X,Y \perp W\vert Z) \Rightarrow (X \perp W\vert Z)$ .
3. $[1 P]$ $(X \perp Y,W\vert Z) \&  (Y \perp W\vert Z) \Rightarrow (X,W\perp Y \vert Z)$ .
4. $$ [1* P] $(X \perp Y \vert Z) \&  (X \perp Y \vert W) \Rightarrow (X \perp Y \vert Z,W)$ .

b)

[1 P]

Figure: Bayesian network.

Apply d-seperation to determine whether the following conditional independence statements hold for the Bayesian network illustrated in Fig. 2.

1. $(X3 \perp X7)$
2. $(X3 \perp X7 \vert X4)$
3. $(X3 \perp X5)$
4. $(X3 \perp X5 \vert X7 )$