Conditional Independence

a)

[1 P] For the probability distribution $P(A,B,C,D)$ with the factorization

$$P(A,B,C,D) = P(A)P(B)P(C\vert A,B)P(D\vert C)$$

show that the following conditional independence assumptions hold.

(i)

$A \perp B \vert \emptyset$

(ii)

$A \perp D \vert C$

b)

[1 P] For the probability distribution $P(A,B,C,D)$ with the factorization

$$P(A,B,C,D) = P(A)P(B\vert A)P(C\vert A)P(D\vert C)$$

show that the following conditional independence assumptions hold.

(i)

$A \perp D \vert C$

(ii)

$B \perp D \vert C$

c)

[1* P] Find a probability distribution $P(A,B,C)$ in form of a probability table which fulfills $A \perp B \vert \emptyset$ but not $A \perp B \vert C$ . Proof that your probability distribution really fulfills the two criteria.