Posted: July 7th, 2022

Problem 1: Descendants of Effects

Problem 1: Descendants of Effects
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 Question Assignment 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.
2
(a) We can describe all guaranteed independences in the Bayes net by defining two or more subsets
of nodes Si
, such that all nodes in Si are independent of all nodes in Sj for i ̸= j. For
example, we can define S1 = {Smokes} and S2 = {Influenza, Sore Throat, Fever} given no
observations. Do the same for conditionally independent nodes i) given Influenza, ii) given
Bronchitis, and iii) given both Influenza and Bronchitis. Make sure your answers capture all
guaranteed independences.
(b) Consider using likelihood weighting to solve two queries, one in which Influenza and Smokes
are observed, and one in which Coughing and Wheezing are observed. Explain how the two
cases differ in the distribution of the resulting samples, as well as the weights that are applied
to the samples.
(c) We perform Gibbs sampling and would like to resample the Influenza variable conditioned on
the current sample (+s, +st, −f, −b, +c, −w). Give a minimal analytical expression for the
sampling distribution Pr(Influenza | sample) (or its unnormalized form). What is the maximum
size of the table that has to be constructed?
(d) Numerically solve for the sampling distribution Pr(Influenza | sample) 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.
Problem 3: Bayes Net 2
The following Bayes net is the “Simple Diagnostic Example” from the Sample Problems of the
Belief and Decision Networks tool on AIspace. All variables are binary.

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Problem 1: Effects’ Descendants
When an arbitrary descendent of a common effect is seen, we will look into the lack of conditional independence guarantees between two random variables. We’ll look at a simple example.

The following is an example of a causal chain of descendants:

Assume that all random variables are of the binary type. A and B’s marginal distributions are identical.

The CPTs of the common effect D0 and its descendants are as follows: uniform (0.5, 0.5), and the CPTs of the common effect D0 and its descendants are as follows:

Pr(+d0 | A, B) A B

1.0 +a +b

+a b 0.5+a b 0.5+a b 0.5+a

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

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