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(1
−ˁ
j,m
)
Q
j,m
(1
−ˁ
j,m
)
Q
j,m
+
ˁ
j,m
,
ʷ
(
−j
)
denotes
all the second-layer indicator vectors except
ʷ
(
j
)
,and
ʲ
(
−j
)
denote all the coefficient
vectors except
ʲ
(
j
)
. So based on Metropolis-Hastings acceptance-rejection rule, we
accept the proposed samples,
ʴ
j
=1and (
ʲ
(
j
)
,ʷ
(
j
)
)=(
ʲ
(
j
)
, ʷ
(
j
)
), with probability
P
add
=min
{
1
, A
j
}
where
p
j,m
=
P
(
ʷ
j,m
=1
|
R
j,m
,ʴ
j
=1
,˃
)=
. Otherwise if the variable
X
j
is active already, that is
ʴ
j
=1,then
based on the current
ʲ
j,m
and
ʷ
j,m
,
m
=1
, ···,M
,wehave
D
j
(
ʘ
j
ₒ
ʘ
j
)
=
P
(
ʘ
j
)
T
(
ʘ
j
ₒ
ʘ
j
)
P
(
ʘ
j
)
·
T
(
ʘ
j
ₒ
ʘ
j
)
⊡
⊛
⊛
M
˃
2
+
˄
j,m
X
j
X
j
2
˄
j,m
˃
2
R
j,m
X
j
˃
2
p
j,m
˄
j,m
˃
j,m
·
⊣
⊝
⊝
ʲ
2
ʲ
j,m
=
ˁ
j,m
·
exp
j,m
−
−
1
m
=1
ʷ
j,m
⊤
⊦
×
1
(1
−ʷ
j,m
)
M
(
ʲ
j,m
−
r
j,m
)
2
2
˃
∗
2
j,m
−
p
j,m
ˁ
j,m
ʸ
j
1
− ʸ
j
−
·
.
(17)
m
=1
Thus the probability of accepting the proposal to remove the variable
X
j
from
S
is
P
del
=min
1
, D
j
}
.
The modified algorithm is shown in Algorithm 3. As mentioned before, in this algo-
rithm, component-wise Gibbs sampler is used to generate the proposal samples of
ʷ
j,m
and
ʲ
j,m
for the corresponding response
Y
m
individually.
{
Algorithm 3: Sample Version of Two-Layer Gibbs Sampler for Union 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. If
ʴ
j
=0,sample
···
(
ʷ
j,m
, ʲ
j,m
)
,m
=1
,
{
···
,M
}
based on the component-wise
Gibbs sample (Step 3 in Algorithm 2) through
R
j,m
. Compute
A
j
in Eq. (16).
1
, A
j
}
Switch
ʴ
j
from 0 to 1 with probability
P
add
=min
{
.If
ʴ
j
=1,set
ʷ
j,m
=
ʷ
j,m
,
ʲ
j,m
=
ʲ
j,m
,
m
=1
,
,M
.
3. If
ʴ
j
=1, suppose the current coefficients and indicators in the second layer are
ʲ
j,m
=
ʲ
j,m
and
ʷ
j,m
=
ʷ
j,m
,
m
=1
,
···
,M
. Compute
D
j
in Eq. (17). Change
ʴ
j
from 1 to 0 with the probability
P
del
=min
···
1
, D
j
}
. If the proposal is rejected,
it means the variable
X
j
is kept in the union model. Then we can re-sample
ʷ
j,m
and
ʲ
j,m
,
m
=1
,
{
,M
for each individual regression model according to the
component-wise Gibbs sampler by
R
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.
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