Biomedical Engineering Reference
In-Depth Information
(a) If Wl
l
= 0; sample
l
from the prior E().
(b) If Wl
l
> 0, sample
l
from N(E
l
;W
l
)1
(
l
>d
l
)
, where
X
n
o
z
i
0
X
l
0
6=l
E
l
= W
1
i
b
l
(t
i
)
l
0
b
l
0
(t
i
) x
0
i
;
l
i=1
z
i
P
l
0
6=l
l
0
fb
l
0
(R
i
) b
l
0
(L
i
)g
b
l
(R
i
) b
l
(L
i
)
d
l
= max(c
l
; 0) and c
l
=
max
i:
i2
=1
:
+
P
i=1
i
x
i
x
0
i
)
1
4. Sample from N(
~
;
~
), where
~
= (
1
0
and
X
~
=
~
0
0
+
1
i
(z
i
(t
i
))x
i
:
i=1
5. Sample from G(a
+ k; b
+
P
l=1
l
):
6. Sample
i
using the rejection sampling as in Holmes and Held (2006).
Algorithm 6.1 is a modification of the Gibbs sampler proposed under the
Probit model with subject-specific variances in the normal densities. The re-
jection sampling method of Holmes and Held (2006) is ecient in sampling
i
in most cases but may encounter numerical problems in some extreme cases.
Also, the rejection sampling method is quite complicated and is not intuitive.
Wang and Lin (2011) proposed a second algorithm that avoids using the
rejection sampling method for sampling
i
. It takes advantage of the facts
that a logistic distribution can be accurately approximated by a student t
distribution with the degrees of freedom 8 (denoted as t
8
) up to a scale con-
stant (Albert and Chib, 1993) and that any t distribution can be regarded as
a scale-normal mixture. Algorithm 6.2 specifically is based on the following
data augmentation:
z
i
= (t
i
) + x
0
i
+
i
a
with constraint zi
i
2 C
i
i
N(0;
i
);
i
Ga(4; 4);
where a = 0:634 is the scale constant and G(4; 4) denotes the Gamma distri-
bution with both shape and rate parameters equal to 4. Integrating out
1
i
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