Information Technology Reference
In-Depth Information
computational cost, instead of deciding whether the j -th variable, X j , is selected or not
based on the posterior probability in Eq. (12) directly, we adopt another method as be-
low. If the current variable is not selected in the union of the support sets, i.e., ʴ j =0,
we propose to active this variable first by setting ʴ j =1, and sample the individual in-
dicators ʷ j,m and coefficients ʲ j,m from the corresponding conditional distributions via
the component-wise Gibbs sampling approach in Chen et al. (2011) [2], i.e. the Step 3
in Algorithm 2. We then decide whether to keep the sampled indicators and coefficients
via the Metropolis-Hastings acceptance-rejection rule. Conversely, if the variable is se-
lected in S ,i.e. ʴ j =1, we then propose to turn down this indicator by switching ʴ j to 0,
and setting all the corresponding indicators ʷ j,m and coefficients ʲ j,m to be zero. There-
fore, we determine whether to accept this proposal or not via the Metropolis-Hastings
acceptance-rejection rule, too. Thus this proposed method can be treated as the sample
version of the two-layer Gibbs sampler. The details of these stages are shown in the
following.
Let ʘ j =( ʴ j ( j ) ( j ) ) be the parameter set for the j th variable, X j ,where ʷ ( j ) =
( ʷ j, 1 ,
j,M ),and ʲ ( j ) =( ʲ j,m ,
j,M ) are the corresponding second-layer in-
dicators and coefficients. The proposed transition of ʘ j can be defined as
···
···
ʘ j )= P ( ʲ ( j ) , ʷ ( j )
T ( ʘ j
R ( j ) j =1 )
|
(14)
T ( ʘ j
ʘ j )=1 ,
(15)
=( ʴ j =1 , ʷ ( j ) , ʲ ( j ) ), R ( j )
=( ʴ j =0 ( j )
= 0 ( j )
where ʘ j
= 0 ), ʘ j
=
ʲ ( j ) , ʷ ( j )
( R j, 1 ,
···
,R j,M ),and
{
}
are sampled from the joint posterior distribution.
Here T ( ʘ j
ʘ j ) is the proposal distribution for changing ʴ j
from 0 to 1, and
T ( ʘ j
ʘ j ) is the proposal distribution to switch ʴ j to 0. Suppose the variable X j is
not included in S currently, i.e. ʴ j =0. Then after sampling ʷ j,m and ʲ j,m by setting
ʴ j =1, we calculate the acceptance probability A j as:
A j ( ʘ j
ʘ j )
= P ( ʘ j )
T ( ʘ j
ʘ j )
P ( ʘ j ) ·
T ( ʘ j
ʘ j )
P ( ʴ j =1 , ʷ ( j ) , ʲ ( j )
| Y −j ( −j ) ( −j ) )
P ( ʴ j =0 ( j ) = 0 ( j ) = 0
1
=
−j ( −j ) ( −j ) ) ·
P ( ʲ ( j ) , ʷ ( j )
| Y
R ( j ) j =1 )
|
= M
ʲ j,m , ʷ j,m j =1 −j −j,m ) P ( ʲ j,m , ʷ j,m |
P ( Y m |
ʴ j =1)
1
ʸ j
×
P ( Y m j =0 −j −j,m )
ʸ j
m =1
1
×
m =1 P ( ʲ j,m |
ʷ j,m ,R j,m j =1 ) P ( ʷ j,m |
R j,m j =1 )
1
exp
M
˃ j,m
˄ j,m ·
˃ 2 + ˄ j,m X j X j
2 ˄ j,m ˃ 2
ʲ j,m + R j,m X j
˃ 2
ˁ j,m
p j,m
ʲ j,m
=
·
m =1
ʷ j,m
×
ˁ j,m
1
(1 −ʷ j,m )
+ ( ʲ j,m
M
r j,m ) 2
2 ˃ 2
j,m
ʸ j
1
·
(16)
p j,m
ʸ j
m =1
 
Search WWH ::




Custom Search