## Computer Science – Bayes Network

Problem 1: Descendants of Effects

We will investigate the absence of conditional independence guarantees between two random vari-

ables 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.

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 pa-

rameters 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.