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MASS
MCAL
Noisy CPT
I
1
I
1
Noisy CPT
I
2
I
2
MASS
MCAL
false
true
false
true
I
1
I
2
false
1
0
false
1
0
true
0.15
0.85
true
0.4
0.6
BC
BC
I
1
I
2
false
true
false
false
1
0
false
true
0
1
true
false
0
1
true
true
0
1
Fig. 9.
Example of a noisy-OR model for breast cancer prediction
patient having
breast cancer
(BC), the so-called effect variable. It is also known
that masses are more frequent occurring sign, which is reflected in the noisy
probability distributions
P
(
I
1
|
MCAL). Finally, the domain
knowledge suggests that the combined occurrence of both signs increases the
probability for having breast cancer, which is modelled by the logical OR as an
interaction function
f
(
I
1
,I
2
) for the effect BC.
Then the probability of having breast cancer given the states of MASS and
MCAL is computed as follows:
MASS) and
P
(
I
2
|
P
(BC =
true
|
MASS
,
MCAL) =
=
P
(BC =
true
|
I
1
,I
2
)
P
(
I
1
|
MASS)
P
(
I
2
|
MCAL)
f
(
I
1
,I
2
)=
true
=
P
(
I
1
=
true
|
MASS)
P
(
I
2
=
true
|
MCAL) +
P
(
I
1
=
true
|
MASS)
P
(
I
2
=
false
|
MCAL) +
P
(
I
1
=
false
|
MASS)
P
(
I
2
=
true
|
MCAL)
.
For example, given the evidence of MASS =
true
and MCAL =
false
then
the probability for breast cancer is:
P
(BC =
true
|
MASS
,
MCAL) = 0
.
85
·
0+0
.
85
·
1+0
.
15
·
0=0
.
85
.
Now suppose that MASS =
true
and MCAL =
true
. Then we obtain
P
(BC =
true
|
MASS
,
MCAL) = 0
.
85
·
0
.
6+0
.
85
·
0
.
4+0
.
15
·
0
.
6=0
.
94
,
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