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Tr ue Bayes boundary
Estimated Bayes bounday
P
(
x
j
)
f
(
x
i
)
f
(
x
j
)
P
(
x
i
)
x
b
x
*
b
Figure 7.2
Error regions associated with approximating the
a posteriori
probability.
same
a posteriori
probability for classes
i
and
j
, that is,
P
(
x
∗
|
i)
=
P
(
x
∗
|
j)
(7.6)
f
i
(
x
b
)
=
f
j
(
x
b
)
(7.7)
Tumer and Ghosh [34] proved that classification error is proportional to the
boundary error
b
x
b
x
∗
. According to Equation 7.6,
=
−
f
i
(
x
∗
+
b)
=
f
j
(
x
∗
+
b)
(7.8)
Tumer and Ghosh [34, 35] showed that the output of any Bayes classifier can
be decomposed into the true Bayes output plus error, that is,
f
c
(
x
)
=
P
(
x
|
c)
+
β
c
+
η
c
(
x
)
(7.9)
where
β
c
and
η
c
are the
bias
and
variance
of the hypothesis.
According to Equation 7.9, Equation 7.8 can be rewritten as
P
(
x
∗
+
b
|
i)
+
β
i
+
η
i
(
x
b
)
=
P
(
x
∗
+
b
|
j)
+
β
j
+
η
j
(
x
b
)
(7.10)
Equation 7.10 can be further approximated by Taylor theorem, that is,
P
(
x
∗
|
i)
+
b
×
P
(
x
∗
|
i)
+
β
i
+
η
i
(
x
b
)
P
(
x
∗
|
j)
+
b
×
P
(
x
∗
|
j)
+
β
j
+
η
j
(
x
b
)
(7.11)
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