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Where
( )=0; In
practical computation, equation (6.27) can be calculated similar to equation
(6.20).
In fact, equation (6.27) is a perception function with a sigmoid activation
function. The input of this function is the possible values of all features. So, to
some extent, naïve Bayesian classifier is equal to a perception model. Further
research demonstrated that naïve Bayesian classifier can be generalized to logical
regression with numerical features.
Consider equation (6.20). If
I
is a characteristic function. if is true, then
I
( )=1; else
I
A j takes discrete values,
count A
(
=
a
C
=
c
)
j
j
i
can be calculated directly from training samples. If
is continuous, it should be
discretized. In unsupervised discretization, a feature is discretized into M equally
wide sections, where
A j
=10 commonly. We can also utilize more complicated
discretization method, such as supervised discretization method.
Let each
M
be a numerical feature (discrete or continuous). The logical
regression model is:
A j
?
P
(
C
=
1
|
A
=
a
,
A
=
a
)
k
à =
1
1
k
k
log
=
b
a
+
b
(6.28)
j
j
0
P
(
C
=
0
|
A
=
a
,
?
A
=
a
)
k
k
j
1
1
1
After transforming as that of equation (6.24), we have:
1
p
=
(6.29)
k
Ã
b a
b
j
j
0
1
+
e
j
=
1
Obviously, this is also a perception function with a sigmoid activation function.
Its inputs are all the feature values. Using function
f j ( ) to replace
b j a j . If the
[
c
,
c
]
sphere of A j is divided into
M
parts and the ith part is
, the function
j i
(
1)
ji
f
j ( ) is:
Ã
M
b a
=
f
(
a
)
=
b I c
(
<
a
c
)
(6.30)
j
j
j
j
ji
j i
(
1)
j
ji
i
=
1
where
b ji is a constant. According to equation (6.29) and (6.30), we have:
1
P C x
(
( )
=
1)
=
(6.31)
k
M
à Ã
(
b I c
(
<
a
c
))
b
1
+
e
ji
j i
(
1)
j
ji
0
j
=
1
i
This is the final regression function. So naïve Bayesian classifier is a
non-parametric and non-linear extension of logical regression. By setting
b
=
(
c
+
c
) /(2
b
)
, we can get a standard logical regression formula.
ji
j i
(
1)
ij
j
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