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influence of Sex to the College Plan is conveyed by the influence of Parental
Encouragement. Besides, the only difference between the two graphs is the
direction of arc between PE and IQ. Two different casual relationships seem both
reasonable. The right network is selected by Sprites et al. in 1993 with
non-Bayesian method.
The most questionable result is that socioeconomic status has direct influence
to IQ. To verify the result, we consider a novel model, which replaces the direct
influence in original model with a hidden variable pointing to SES and IQ.
Besides, we also consider such models where hidden variable points to SES, IQ
and PE and none or one or both of two links of SES-PE and PE-IQ are removed.
For each structure, the number of hidden variables in these models varies from 2
to 6.
H
SEX
IQ
PE
SES
CP
Log
p
(
D
|
S
)
-45629
Figure 6.2. The posteriori most likely network structure with a hidden variable
We use the Cheeseman-Stutz variant of Laplace Approximation to compute
the posterior probabilities of these models. To find the MAP
θ
!
, we use EM
algorithm and take the largest local maximum from 100 runs with different
random initial
θ
s
!
. The model with the highest MAP is shown in Figure 6.2. This
model is 2×10
10
times more likely than the best model containing no hidden
variables about. Another most likely model contains a hidden variable and has an
additional arc from the hidden variable to PE. This model is only 5×10
-9
times
less likely than the best model. Suppose that no reasonable models are ignored,
strong evidence suggests that there is a hidden variable influencing the SES and
IQ. An examination of the probabilities in Figure 6.2 suggests that the hidden
variable corresponds to some concept like “parent quality”.
s
