Page 280 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

distribution. The size of j is determined rather arbitrarily, but a graphical illustration
could help.
Several strategies can be implemented in using the Gibbs sampler and these have
an effect on the degree of correlation between the values sampled. Details of various
strategies are discussed in detail by Sorensen and Gianola (2002) and therefore not
2
2
presented here. One approach is to run a single long chain. A sample (b, u, s , s ) is
u e
saved at every dth iterate until a total t samples are saved and analysed. The larger d is,
the lower the degree of autocorrelation between the samples. Another strategy, known
as the multiple chain or short chain approach, involves carrying out several parallel
t runs and saving the last nth sample from each run. Thus this approach produces
m = nt samples. The different chains will produce different realizations, even if the
same starting values are used. However, if the parameters in the model are highly
correlated, it might be useful to utilize different starting values in the different chains.
Determining convergence with the Gibbs sampler is not very straightforward, but
it is advisable, depending on the size of the problem, to run several chains and check
convergence graphically.

16.2.3 Inferences from the Gibbs sampling output

The samples saved are usually analysed to estimate posterior means or variances of
the posterior distribution. Detailed discussion of various estimation methods is given
in Sorensen and Gianola (2002) and not presented here. Given that w is a vector of
size k, containing the saved samples, then the posterior mean and variance can be
computed, respectively, as:
k
å fw )
(
i
m = i=1 (16.15)
f
k
and:
k
å (( fw - f ) 2
) m
i
var(m ) = i=1
f
k
where f(w) is a function of interest of the variables in w. For instance, in the linear
animal model in Section 6.2, the function of interest would be the variance components
2
2
(s and s ) and the vectors b and u.
u e
The above estimates from the posterior distribution are associated with sampling
variance (Monte Carlo variance). The larger the number of samples analysed, the
smaller the sampling variance. It is usually useful to get an estimate of the sampling
variance associated with the estimates from the posterior distributions. An empirical
estimate could be obtained by running several independent runs and then computing
the between-chain variance of the estimates obtained for each run. This is not com-
putationally feasible in most practical situations and various methods are used to
estimate this variance. A number of such estimators are fully discussed by Sorensen
and Gianola (2002). A simple method that could be used involves calculating the

264 Chapter 16