We will investigate the absence of conditional independence guarantees between two random variables when an arbitrary descendant of a common effect is observed. We will consider the simple
case of a causal chain of descendants:
Suppose that all random variables are binary. The marginal distributions of A and B are both
uniform (0.5, 0.5), and the CPTs of the common effect D0 and its descendants are as follows:
A B Pr(+d0 | A, B)
+a +b 1.0
+a −b 0.5
−a +b 0.5
−a −b 0.0
Di−1 Pr(+di
| Di−1)
+di−1 1.0
−di−1 0.0
(a) Give an analytical expression for the joint distribution Pr(D0, D1, · · · , Dn). Your expression
should only contain CPTs from the Bayes net parameters. What is the size of the full joint
distribution, and how many entries are nonzero?
(b) Suppose we observe Dn = +dn. Numerically compute the CPT Pr(+dn|D0). Please show how
you can solve for it using the joint distribution in (a), even if you do not actually use it.
(c) Let’s turn our attention to A and B. Give a minimal analytical expression for Pr(A, B, D0, +dn).
Your expression should only contain CPTs from the Bayes net parameters or the CPT you found
in part (b) above.
(d) Lastly, compute Pr(A, B | +dn). Show that A and B are not independent conditioned on Dn
