Consider the following learning problem in which we wish to c
lassify inputs, each consisting of a
single real number, into one of two possible classes
C
1
and
C
2
. There are three potential hypotheses
where
Pr(
h
1
) = 3
/
10
,
Pr(
h
2
) = 5
/
10
and
Pr(
h
3
) = 2
/
10
. The hypotheses are the following
functions
h
i
(
x
) =
x

i

1
5
and the likelihood for any hypothesis
h
i
is
Pr(
x

C
1
|
h
i
, x
) =
σ
(
h
i
(
x
))
where
σ
(
y
) = 1
/
(1 + exp(

y
))
. You have seen three examples:
(0
.
9
, C
1
)
,
(0
.
95
, C
2
)
and
(1
.
3
, C
2
)
,
and you now wish to classify the new point
x
= 1
.
1
.
1. Explain how in general the
maximum a posteriori (MAP)
classifier works. [3 marks]
2. Compute the class that the MAP classifier would predict in t
his case. [10 marks]
3. The preferred alternative to the MAP classifier is the Baye
sian classifier, computing
Pr(
x

C
1
|
x,
s
)
. where
s
is the vector of examples. Show that
Pr(
x

C
1
|
x,
s
) =
X
h
i
Pr(
x

C
1
|
h
i
, x
)Pr(
h
i
|
s
)
What are you assuming about independence in deriving this re
sult? [3 marks]
4. Compute the class that the Bayesian classifier would predi
ct in this case. [4 marks