1
Introduction to Probabilistic Graphical Models
December 26, 2006
Problem set 4  Solutions
Lecturer: Eran Segal
1.
(a) This can be done by summing out the variables that we wish to remove and incorpo
rating their effects into the leak noise parameter. The important thing to see here is
that the summing out can be done efficiently.
P
(
F
i
=
f
0
i
, D
1
, ..., D
‘
)
=
X
D
‘
+1
,...,D
k
P
(
F
i
=
f
0
i
, D
1
. . . D
k
)
=
X
D
‘
+1
,...,D
k
P
(
F
i
=
f
0
i

D
1
...D
k
)
·
P
(
D
1
. . . D
k
)
=
X
D
‘
+1
,...,D
k
P
(
F
i
=
f
0
i

D
1
...D
k
)
·
k
Y
i
=1
P
(
D
i
)
=
X
D
‘
+1
,...,D
k
(1

λ
0
)
k
Y
j
=1
(1

λ
j
)
d
j
·
k
Y
i
=1
P
(
D
i
)
=
(1

λ
0
)
‘
Y
j
=1
(1

λ
j
)
d
j
P
(
D
j
)
X
D
‘
+1
,...,D
k
k
Y
i
=
‘
+1
(1

λ
i
)
d
i
·
P
(
D
i
)
For each
D
i
where
i > ‘
the term in parentheses can be simplified as follows:
(1

λ
i
)
P
(
D
i
= 1) +
P
(
D
i
= 0) = 1

λ
i
P
(
D
i
= 1)
And so the entire term in parentheses can be written as:
k
Y
i
=
‘
+1
(1

λ
i
P
(
D
i
= 1))
We will denote this term as
A
.
Notice that A can be computed efficiently in the price of
O
(
k
) instead of a trivial
summing out in
F
i
’s CPD, in the price of
O
(2
k
).
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 Fall '07
 EranSegal
 D1, DI, dk

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