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Once
X
j
is not selected, then we can simply set
ʷ
j,m
=0and
ʲ
j,m
=0for all
m
=1
,...,M
.Otherwise,ifthevariable
X
j
is included in
S
,i.e.
ʴ
j
=1, then we need
to check if
X
j
is active or not for each individual response
Y
m
separately. Following
the component-wise Gibbs sampler in Chen et al. (2011) [2], the likelihood ratio
Q
j,m
of the variable
X
j
with respect to the
m
-th model,
Y
m
, is computed and can be shown
as
Q
j,m
=
P
(
Y
m
|
ʷ
j,m
=1
,ʷ
−j,m
,ʲ
−j,m
,˃,ʴ
j
=1)
P
(
Y
m
|
ʷ
j,m
=0
,ʷ
−j,m
,ʲ
−j,m
,˃,ʴ
j
=1)
(
R
j,m
X
j
)
2
˄
j,m
2
˃
2
(
˃
2
+
X
j
X
j
˄
j,m
)
˃
X
j
X
j
˄
j,m
+
˃
2
·
=
exp
j,m
/˄
j,m
exp
r
j,m
.
=
˃
2
(11)
2
˃
2
j,m
Based on both likelihood ratio functions, Eq. (10) and Eq. (11), the corresponding
posterior probabilities of
ʴ
j
=1and
ʷ
j,m
=1can be derived. The proposed Gibbs
sampling algorithm is summarized in Algorithm 2. Note that we would start from the
null model by setting
ʴ
j
=0;
ʷ
j,m
=0and
ʲ
j,m
=0for all
j
and
m
. Based on our
experiences, this initial model works well.
Algorithm 2: The Two-Layer Gibbs Sampler for Support Recovery
1. Randomly select a variable
X
j
. Compute
R
j,m
=
Y
m
−
i
=
j
X
i
ʲ
i,m
for
m
=
1
,
,M
.
2. Compute the likelihood ratio
Z
j
according to Eq. (10), and then evaluate the poste-
rior probability of
ʴ
j
···
(1
− ʸ
j
)
Z
j
P
(
ʴ
j
=1
|
Y
,ʴ
−j
,
{
ʲ
−j,m
,m
=1
,
···
,M
}
,˃
)=
.
(12)
(1
−
ʸ
j
)
Z
j
+
ʸ
j
3. Sample
ʴ
j
based on the posterior probability in Eq. (12). If
ʴ
j
=0,thenset
ʷ
j,m
=
0 and
ʲ
j,m
=0,forall
m
=1
,
,M
,
compute the likelihood ratio
Q
j,m
according to Eq. (11), and sample
ʷ
j,m
based on
the posterior probability
···
,M
. Otherwise, for each
m
=1
,
···
(1
−
ˁ
j,m
)
Q
j,m
P
(
ʷ
j,m
=1
|
Y
m
,ʷ
−j,m
,ʲ
−j,m
,˃
)=
.
(13)
(1
−
ˁ
j,m
)
Q
j,m
+
ˁ
j,m
If
ʷ
j,m
=0,set
ʲ
j,m
=0; otherwise, sample
ʲ
j,m
∼ N
(
r
j,m
,˃
2
j,m
).
4. After repeat above steps for all variables, compute the current residual matrix,
Res
=
,
(
diaq
(
Res
Res
))
/M
+
b
2
IG
(
a
+
n
2
B
. Then sample
˃
2
Y
−
X
∼
).Goto
Step 1.
3.3 Sample Version of Two-Layer Gibbs Sampler
In Algorithm 2, the computation of
Z
j
in Eq. (10) involves 2
M
cases and can be com-
putational expensive, especially when the number of the responses,
M
,islarge.Tosave
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