Biomedical Engineering Reference
In-Depth Information
7.3
Posterior Computation
In this section, we discuss the posterior computation only under the prior
specified by the dynamic model in Section 7.2.6 as the posterior computation
under the other priors is either similar or more straightforward. Again, we use
the generic notation given in Section 7.2.6. Under the dynamic prior, the joint
prior density of '(t) and w 2
is given by
n '( 1 ) 2
2c 0 w 2
o
('(t);w 2 ) / 0
( 0 ) (w 2 ) 0 1 exp 0
(! 2 ) J 2 exp
0
w 2
'( k ) '( k1 ) 2
2w 2
exp n
o :
Y
k2
(7.13)
Each component of has its own variable w 2 . Multiplying p+ 1 such densities
in the form of (7.13) gives the joint density of and w 2 's. The posterior
distribution of , w 2 's, J's, and ( 1 < 2 < ::: < J )'s is proportional to
the product of the likelihood L(jD obs ) in Equation (7.3) and the joint prior
density.
Due to the complexity of the posterior distribution with the dynamic prior
for for interval-censored data D obs , analytical evaluation of the posterior dis-
tribution is not possible. The posterior computation is carried out via MCMC
sampling. As shown in the sequel, treating the even times that are censored
by finite intervals (i.e., Ri i < 1) as latent variables greatly facilitates the
sampling. The MCMC sampling algorithm draws fT i : R i < 1g, , and w 2 's
iteratively:
Step 1. For each finitely censored subject i, sample event time Ti i given .
Step 2. For each component '(t) in , sample the time-varying coecient
function given all other components in , the event time T i 's, w 2 's, and
observed data. A reversible jump algorithm is required because a change
in J leads to a posterior distribution with a different dimension of the
model parameters.
 
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