CS195f Homework 2
Mark Johnson and Erik Sudderth
Homework due at 2pm, 1st October 2009
The frst question asks you to analyse the Following naive Bayes model that describes the
weather in a mythical country.
Y
=
{
night
,
day
}
X
1
=
{
cold
,
hot
}
X
2
=
{
rain
,
dry
}
P(
X
1
,X
2
,Y
) = P(
Y
)P(
X
1

Y
)P(
X
2

Y
)
P(
Y
=
day
) = 0
.
5
P(
X
1
=
hot

Y
=
day
) = 0
.
9
P(
X
1
=
hot

Y
=
night
) = 0
.
2
P(
X
2
=
dry

Y
=
day
) = 0
.
75
P(
X
2
=
dry

Y
=
night
) = 0
.
4
Question 1:
For each of the following formulae except the ±rst, write an equation which de±nes it in
terms of formulae that appear earlier in the list. (For example, you should give a formula
for
P(
x
1
,x
2
)
in terms of
P(
x
1
,x
2
,y
)
). Then given the model above, calculate and write out
the value of the formula for possible each combination of values of the variables that appear
in it.
a)
P(
x
1
,x
2
,y
)
.
b)
P(
x
1
,x
2
)
.
c)
P(
y

x
1
,x
2
)
.
d)
P(
x
1
)
.
e)
P(
x
2
)
.
f)
P(
x
1

x
2
)
.
g)
P(
x
2

x
1
)
.
1
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P(
x
1

x
2
,y
)
.
i)
P(
x
2

x
1
,y
)
.
Are
X
1
and
X
2
conditionally independent given
Y
? Are
X
1
and
X
2
marginally independent,
integrating over
Y
? Provide a short proof for both answers.
Consider a binary categorization problem, and let
p
(
y
i

x
i
) denote the posterior distri
bution of the latent class label
y
i
∈ {
0
,
1
}
given observation
x
i
. Suppose that the classiFer
ˆ
y
(
x
i
) is allowed to make one of three decisions: choose class 0, choose class 1, or “reject”
this data (refuse to make a decision). We can use a Bayesian decision theoretic approach to
tradeo± the losses incurred by incorrect decisions and rejections.
Question 2:
Suppose that the classiFer incurs a loss of
0
whenever it chooses the correct class, a loss of
1
whenever it chooses the wrong class, and a loss of
λ
whenever it selects the reject option.
Express the optimal decision rule
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 Spring '09
 johnson
 Machine Learning, Type I and type II errors, x1 x2, posterior distribution

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