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f
Y|t
(t−e)
f
Y|t
(y)
f
Y|−1
(−1−e)
f
Y|−1
(y)
1
1
f
Y|1
(1−e)
f
Y|1
(y)
y
b
a
d
c
0
0
−1
0
1
−2
−1
0
1
e
2
(a)
(b)
Fig. 2.3 Illustration of the transformation
E
=
T −Y
for a two-class classification
problem, emphasizing the fact that
f
E
(0) = 0
for continuous class-conditionals.
Tabl e 2 . 1
Decision table corresponding to Fig. 2.3b.
Classifier
output
−
1
1
True
class
−
1
a
b
1
c
d
them in terms of
f
Y
(
y
) using the following theorem.
Theorem 2.1.
Suppose
X
1
,X
2
, ..., X
k
are continuous random variables
and
Y
=
g
(
X
1
,X
2
, ..., X
k
)
for some function
g
. Suppose also that
X
k
|
X
1
···
g
(
x
1
,...,x
k
)
|
f
X
1
,...,X
k
(
x
1
,...,x
k
)
dx
1
...dx
k
<
∞
.
(2.23)
Then
[
Y
]=
E
X
1
···
g
(
x
1
,...,x
k
)
f
X
1
,...,X
k
(
x
1
,...,x
k
)
dx
1
...dx
k
.
(2.24)
X
k
For a proof of this theorem see, for instance, [61, 87]. Theorem 2.1 allows
us to compute an expected value of an r.v.
Y
either directly according to
the definition (i.e., in terms of
f
Y
(
)), or whenever
Y
is given by a certain
function of other r.v.'s as in (2.24). The class-conditional expected value
expressed in (2.3) — where
X
may be a multidimensional r.v. — is then
simply
E
Y |t
[
L
(
t, Y
)]. Therefore, assuming
Y
restricted to [
·
−
1
,
1],wemay
write
