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10708 Graphical Models: Homework 3
Due October 27th, beginning of class
October 13, 2006
Instructions
: There are six questions on this assignment. Each question has the name of
one of the TAs beside it, to whom you should direct any inquiries regarding the question.
The last problem involves coding, which should be done in MATLAB. Do
not
attach your
code to the writeup. Instead, copy your implementation to
/afs/andrew.cmu.edu/course/10/708/your_andrew_id/HW3
Refer to the web page for policies regarding collaboration, due dates, and extensions.
Note
: Please put your name and Andrew ID on the ﬁrst page of your writeup.
1
Triangulation
[10 pts] [Khalid]
A
B
C
D
E
F
G
H
Figure 1: Bayes net for question 1
1. Moralize the Bayes net in ﬁgure 1.
2. Supply a perfect elimination ordering (
i.e.
, one that yields no ﬁll edges).
1
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View Full Document3. Supply an elimination ordering that yields a triangulated graph with at least 5 nodes
in one or more cliques
4. Draw clique trees for the elimination orderings in parts 2 and 3.
2
Clique Tree Factorization
[10 pts] [Khalid]
2.1
Prove that the clique beliefs
π
(
C
i
) =
P
(
C
i
) and edge beliefs
μ
ij
(
S
ij
) =
P
(
S
ij
) form a ﬁxed
point for the belief propagation algorithm for a clique tree, i
.
e
.
, if we start BP with these
beliefs, no messages will change them.
2.2
Using the independencies we ask you to show in Question 4, prove that in a clique tree for
a BN we can represent the joint distribution by:
P
(
X
) =
Q
i
P
(
C
i
)
Q
ij
P
(
S
ij
)
.
You should not “prove” by corollary from the correctness of BP in clique trees.
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 Fall '07
 CarlosGustin

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