Biomedical Engineering Reference
In-Depth Information
The augmented likelihood in Equation (6.6) can be regraded as the com-
plete data likelihood, with all z
il
s and w
il
s being missing data. This likelihood
is simply a product of Poisson probability mass functions and forms the basis
of the following Gibbs sampler below.
1. Sample all zi,
i
, z
il
, w
i
, and w
il
.
Let z
i
= 0 and w
i
= 0 for all i, and zi
il
= 0 and w
il
= 0 for all i and l.
If
i1
= 1, then sample
z
i
Pf(R
i
) exp(x
0
i
)g1
(z
i
>0)
;
(z
i1
; ;z
ik
) M(z
i
;p
i
); p
i
/f
1
b
1
(R
i
);:::;
k
b
k
(R
i
)g;
where M denotes a multinomial distribution. If
i2
= 1, then sample
w
i
P[f(R
i
) (L
i
)gexp(x
0
i
)]1
(w
i
>0)
;
(w
i1
; ;w
ik
) M(w
i
; q
i
);
q
i
/ [
1
fb
1
(R
i
) b
1
(L
i
)g;:::;
k
fb
k
(R
i
) b
k
(L
i
)g]:
2. Sample
j
from its full conditional distribution, [
j
j] / (
j
)h(), using
the adaptive rejection sampling (ARS) method (Gilks and Wild, 1992)
for j = 1;:::;p. The full conditional distribution of
j
is log-concave
as long as its prior (
j
) is log-concave. Up to an additive constant
logfh()g is equal to
h
x
0
i
(z
i
i1
+ w
i
i2
) exp (x
0
i
)f(R
i
)(
i1
+
i2
) + (L
i
)
i3
g
i
X
:
i
3. For l = 1;:::;k, sample
l
from Gamma distribution G(aγl
l
;b
l
), where
1 +
X
i
a
l
=
fz
il
i1
1
(b
l
(R
i
)>0)
+ w
il
i2
1
(b
l
(R
i
)b
l
(L
i
)>0)
g;
b
l
= +
X
i
exp(x
0
i
)f(
i1
+
i2
)b
l
(R
i
) +
i3
b
l
(L
i
)g:
4. Sample from G(a
+ k; b
+
P
l
l
).
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