# CS 726 Homework 1 (Due Feb 4, 2010) Write

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```					                                      CS 726: Homework 1 (Due Feb 4, 2010)

Write your answers in the space provided. You are expected to solve each question on your own. Do not try to
search the answers from any external sources, like the web. You are allowed to discuss a few questions with your
classmates provided you mention their names. Also, mention whether you are crediting/auditing/sitting-through
the course.

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1. Let xi and xj be two variables that are not adjacent in a Bayesian network. Show that xi ⊥ xj |P a(xi ), P a(xj ).

2. If a distribution factorizes as follows Pr(x1 , x2 , x3 , x4 , x5 ) = Pr(x1 ) Pr(x2 ) Pr(x3 |x1 ) Pr(x4 |x2 ) Pr(x5 |x3 , x4 ),
⊥
show that x3 ⊥ x4 .

3. Suppose you have a distribution over four binary variables given as follows:

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12     if (x1 ∨ ¬x2 ∨ x3 ) ∧ (x2 ∨ ¬x3 ∨ x4 )
P r(x1 , x2 , x3 , x4 ) =
0      otherwise

Construct the Bayesian network corresponding to this distribution using the variable order x1 , x2 , x3 , x4 .

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4. Consider the Bayesian Network represented by the following graph:

(a) Assume that the vertex H and the dotted edges C → H and H → F are absent. State with reasons
whether the following conditional independence statements hold:
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i. A ⊥ G | F

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ii. A ⊥ D

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iii. B ⊥ G | C, F

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⊥
iv. F ⊥ D | C

(b) Now add the vertex H along with the two dotted edges to the graph. Find the smallest set S (G, H ∈ S)
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such that G ⊥ H | S. Provide a proof if no such S exists.

5. Deﬁne a set Z of nodes to be self-contained if, for every pair of nodes A, B ∈ Z, and any directed trail
between A and B, all nodes along the trail are also in Z.
(a) Consider a self-contained set Z, and let Y be the set of all nodes that are a parent of some node in Z
but are not themselves in Z. Let U be the set of nodes that are an ancestor of some node in Z but that
are not already in Y ∪ Z. Prove, based on the d-separation properties of the network, that Z⊥U | Y.
Make sure that your proof covers all possible cases.

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(b) Provide a counterexample to this result if we retract the assumption that Z is self-contained.

6. Recall the deﬁnition of blocked trails in d-seperation. A trail between two nodes X and Y in a Bayesian
network is active if it is not blocked as per the d-separation test. A minimal active trail between X and Y
is an active trail X = X1 − − − . . . − − − Xi − − − . . . − − − Xk = Y such that no proper subset of nodes
in this trail is active. A triangle in a minimal active trail is a set of three consecutive nodes in the trail
Xi−1 − − − Xi − − − Xi+1 with a direct edge between Xi−1 and Xi+1 .

(a) Prove that the only way in which a triangle can appear in a minimal active trail with an edge between
Xi−1 and Xi+1 is as Xi−1 ← Xi → Xi+1 , that is the other three conﬁgurations are not minimal.

(b) Show that in any such triangle above in a minimal active trail, at least one of Xi−1 or Xi+1 should be
a V-node, that is, the trail can extend as either,
• Xi−1 ← Xi → Xi+1 ← Xi+2 , or
• Xi−2 → Xi−1 ← Xi → Xi+1 .

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