The methods outlined in Section 
are used to
sample from the posterior distribution. In particular Metropolis
Hastings (Section ) and Gibbs sampling
(Section ) are used.
The posterior distribution may be written 
,
where
The distribution of the hyperparameters is independent of the data, conditional on the crack specific parameters, which yields the following result.
and extending the 
notation in the obvious fashion,
The sampling methods as described are used to sample from the posterior distribution
of 
in the following fashion. The actual calculations for the constant
and multiplicative variance are provided in the appendix.