be a random vector with distribution given by the graphical model in
Figure 2. Consider variable
X
1
. What is the minimal subset of the variables,
A
⊆ X {
X
1
}
,
such that
X
1
is independent of the rest of the variables,
X 
A
∪ {
X
1
}
, givan
A
? Justify
your answer.
2
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View Full DocumentFigure 2: Graphical Model for Prob. 2
2.2
[8 pts]
Now, let the distribution of
X
be given by some graphical model instance,
B
= (
G
, P
).
Consider variable
X
i
. What is the minimal subset of the variables,
A
X {
X
i
}
, such that
X
i
is independent of the rest of the variables,
X 
A
∪{
X
i
}
, given
A
? Prove that this subset
is necessary and suﬃcient.
(Hint: Think about the variables that
X
i
cannot possibly be conditionally independent of,
and then think some more)
2.3
Extra Credit [8 pts]
Show how you could eﬃciently compute the distribution over a variable
X
i
given some as
signment to all the other variables in the network:
P
(
X
i

x
1
, ..., x
i

1
, x
i
+1
, ..., x
n
).
Your procedure should not require the construction of the entire joint distribution
P
(
X
1
, ..., X
n
).
Specify the computational complexity of your procedure.
3
Marginalization [15 pts]
1. Consider the Burglar Alarm network shown in Figure 3. Construct a Bayesian network
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 Spring '13
 Dr.ZAre
 Probability theory, pts, 3 pts, 4 pts, graphical model

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