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.
Problem 2: Bayes Net 1
The following Bayes net is the “Fire Alarm Belief Network” from the Sample Problems of the Belief
and Decision Networks tool on AIspace. All variables are binary.
(a) Which pair(s) of nodes are guaranteed to be independent given no observations in the Bayes
net? Now suppose Alarm is observed. Identify and briefly explain the nodes whose conditional
independence guarantees, given Alarm, are different from their independence guarantees, given
no observations.
(b) We are interested in computing the conditional distribution Pr(Smoke | report). Give an
analytical expression in terms of the Bayes net CPTs that computes this distribution (or its
unnormalized version). What is the maximum size of the resultant table if all marginalization
is done at the end?
(c) We employ variable elimination to solve for the query above. Identify a variable ordering that i)
yields the greatest number of operations possible, and ii) yields the fewest number of operations
possible. Also give the max table sizes in each case.
(d) Following your second variable ordering above, numerically solve for Pr(Smoke | report) using
the default parameters in the applet example. You may check your answer using the applet,
but you should work it out yourself and show your work.
