Information Technology Reference
In-Depth Information
4
Simulated Example
In this section, we illustrate the performance of the proposed two-layer Gibbs sampler
via a simulated example. In this example, we set
M
=15and there are
p
= 200
predictor variables of length
n
=80. The variables are defined by
X
j
=
G
j
+
kG,
where
k
=2is a pre-specified constant, and
G
j
's and
G
are independently generated
from multivariate normal distribution with zero mean vector and identical covariance
matrix
I
80
. In the setting, the correlation between any two variables is 0.8. The true
active variable set was
,and
the corresponding coefficients of 15 single regression models are shown in Table 1. The
other coefficients are all set to be zero. Then each response is generated according to
the linear model
Y
m
=
{X
7
,X
8
,X
9
,X
11
,X
12
,X
13
}
,i.e.
S
=
{
7
,
8
,
9
,
11
,
12
,
13
}
N
80
(
0
,I
80
).
The sample version of two-layer Gibbs sampler, Algorithm 3, is used in this ex-
ample. The prior parameters are set as
ʸ
j
X
ʲ
m
+
ˉ
m
,where
ˉ
m
∼
=0
.
5,
˄
j,m
=0
.
5,
ˁ
j,m
=20for all
j
,and
a
=
b
=0
.
001 as the non-informative
parameter for inverse gamma prior of
˃
2
. The initial model is set as the null model,
i.e.
ʴ
j
=0;
ʷ
j,m
=0and
ʲ
j,m
=0for all
j
and
m
. Totally we run 3000 sweeps.
After discarding the first 2000 sweeps, samples collected from the last 1000 sweeps
are used for the inference about support union recovery. First the posterior probabilities
P
(
ʴ
j
=1
∈{
1
,
···
,
200
}
,m
∈{
1
,
···
,
15
}
ʴ
j
=1
,
Y
) are estimated based on the posterior samples,
and then for the posterior inference, the median probability criterion is used according
to Barbieri and Berger (2004) [1]. Thus the threshold probabilities for including predic-
tor in the shared and individual model are both set to 0.5, i.e.
P
(
ʴ
j
=1
|
Y
) and
P
(
ʷ
j,m
=1
|
|
Y
)
≥
0
.
5 and
P
(
ʷ
j,m
=1
0
.
5.
The posterior probabilities of
P
(
ʴ
j
=1
|
ʴ
j
=1
,
Y
)
≥
) are shown in Figure 1. In fact, the poste-
rior probabilities of
X
7
,X
8
,X
9
,X
11
,X
12
and
X
13
are all higher than 0.5. Therefore,
the selection result of the support union recovery agrees the true model.
Then
P
(
ʷ
j,m
=1
|
Y
S
are shown in Figure 2. For those nonzero
coefficients in Table 1, all corresponding indicators in the second set have posterior
probability larger than 0.5. Therefore, based on the median probability criterion, these
are treated as active. In particular,
ʲ
8
,
6
=0
.
4,
ʲ
9
,
5
=0
.
4,
ʲ
9
,
6
=0
.
4,
ʲ
9
,
9
=0
.
5,
and
ʲ
9
,
14
=0
.
5 are relatively smaller values. However, the corresponding posterior
|
ʴ
j
=1
,
Y
) for
j
∈
Ta b l e 1 .
The true coefficients of the union of the support sets
S
in the simulated example
X
j
ʲ
j,
1
ʲ
j,
2
ʲ
j,
3
ʲ
j,
4
ʲ
j,
5
ʲ
j,
6
ʲ
j,
7
ʲ
j,
8
ʲ
j,
9
ʲ
j,
10
ʲ
j,
11
ʲ
j,
12
ʲ
j,
13
ʲ
j,
14
ʲ
j,
15
X
7
0.9
1.7
0
1.2
0.5
0
2.1 0.7
0
0.8
0.8
2.5
0
0
0.9
X
8
0.9
1.7 2.2 1.2
0
0.4
2.1 0.7
0
0.8
0.8
2.5
1.3
0
0
X
9
0.9
1.7
0
0
0.5 0.4
2.1
0
0.5
0.8
0.8
2.5
0
0.5
0
X
11
0
0
0
0
1.3
0
0
0
0
0
0
0
0
0
0
X
12
0
0
0
0
0
0
0
0
0.7
0
0
0
0
0
0
X
13
0
0.6
0
0
0
0.5
0
0
0
0
0
0
0
0
0
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