Thus the posterior distribution is 
where
and
, thus in this case the posterior
distribution for 
is 
.
The posterior distribution for the crack specific parameters may
be obtained in the same fashion as for the growth model, and are
sampled using MCMC. The additional pair of parameters which are
sampled for each crack comprise 
and 
. These are sampled
in a similar fashion to the other crack specific parameters since
is a deterministic function of these
values. Thus 
which is sampled in the same fashion as the other parameters for
the growth model.