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1
Introduction to Probabilistic Graphical Models
December 13, 2007
Problem set 3  Solutions
Lecturer: Eran Segal
1. (a) Prove the following theorem: Let
G
1
and
G
2
be two graphs over
X
. If they have the
same skeleton and the same set of vstructures then they are Iequivalent.
Answer:
Assume that
G
1
and
G
2
have the same skeleton and the same vstructures.
First we assume that
(
X
⊥
Y

Z
)
∈
I
(
G
1
)
and we show that
(
X
⊥
Y

Z
)
∈
I
(
G
2
)
.
By saying that the two graphs have the same skeleton we say that they have the same
trails. Lets look on some trail between
X
∈
X
and
Y
∈
Y
in
G
1
that given
Z
is inactive,
and we will show that this trail in
G
2
is inactive too. Consider two cases:
i. The trail in
G
1
is inactive because some (at least one) of the nodes on the trail that
are not in a v–structure are observed (in
Z
). Then clearly these nodes also blocks
the trail in
G
2
.
ii. Otherwise, all nodes on the trail that are not in a v–structure are not observed (not in
Z
), but then for some vstructure
V
i

1
,V
i
,V
i
+1
on the trail, non of the descendents
of
V
i
(including
V
i
itself) are observed. That is for every node
V
such that there
is a directed path in
G
1
from
V
i
to
V
, all the nodes on the path are not observed.
Consider such a directed path from such a
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 Fall '07
 EranSegal

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