2. (
X
⊥
Y

Z
) and (
X, Y
⊥
W

Z
) imply (
X
⊥
W

Z
).
1
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3. (
X
⊥
Y, W

Z
) and (
Y
⊥
W

Z
) imply (
X, W
⊥
Y

Z
).
1.4
[3 pts]
Provide an example of a distribution
P
(
X
1
, X
2
, X
3
) where for each
i
̸
=
j
, we have that
(
X
i
⊥
X
j
)
∈
I
(
P
), but we also have that (
X
1
, X
2
⊥
X
3)
/
∈
I
(
P
).
1.5
[8 pts]
Figure 1: Graphical Model for Prob. 1.5
Let
X
,
Y
,
Z
be binary random variables with joint distribution given by the graphical model
shown above (
v
structure). We define the following shorthands:
a
△
=
P
(
X
=
t
);
b
△
=
P
(
X
=
t

Z
=
t
);
c
△
=
P
(
X
=
t,

Z
=
t, Y
=
t
)
1. For all the following cases, provide examples of conditional probability tables (CPTs)
(and compute the quantities,
a
,
b
,
c
), which make the statements true:
(a)
a > c
(b)
a < c < b
(c)
b < a < c
2. Think of
X
,
Y
as causes and
Z
as a common effect, and for all the above cases
summarize (in a sentence or two) why the statements are true for your examples.
(Hint: Think about positive and negative correlations along edges)
2
Graph independencies [12pt]
2.1
[4 pts]
Let
X
=
{
X
1
, ..., X
n
}
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.
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 Spring '13
 Dr.ZAre
 Probability theory, pts, 3 pts, 4 pts, graphical model

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