Page 285 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Including the prior distribution, the above can be expressed (Jensen et al., 1994) as:
é ù
P(| b,u,y μ | R | - 1 2 ( v e + p+ + m) exp - ê 1 tr {R -1 (S + V e -1 )}ú
2
1
)
R
e
ë ê 2 û ú
2
where m is the number of records and S is:
e
′
′
ˆˆ
æ ˆˆ ee ö
ee
S = ç 1 1 1 2 ÷
2
e
′
ˆˆ
ˆˆ
ee
è 2 ′ 1 ee 2 ø
2
ˆ
assuming that n = 2 and e = y − X b − Z u , i = 1, n.
i i i i i i
Thus:
−1 −1
2
R| b, u, y ~ IW ((S + V ) , v + m) (16.25)
2 e e e
which is in the form of a p-dimensional inverted Wishart distribution with v + m
e
−1
2
degrees of freedom and scale parameter (S + V ).
e e
Similarly, the conditional distribution for the additive genetic variance is:
P(G|b,u,y) ∝ P(G)P(u|G)
Including the prior distribution, the above can be expressed (Jensen et al.,
1994) as:
⎡ ⎤
P(|b,u,y ∝ | G | − 1 2 (v u + + +q ) exp − ⎢ 1 tr {G −1 (S + V u −1 )}⎥
2
p
1
G
)
u
⎣ ⎢ 2 ⎦ ⎥
2
where q is the number of animals and, assuming n = 2, S is:
u
æ ′ -1 u A u ö
′
-1
uA u
S = ç 1 1 1 2 ÷
2
u ç -1 -1 ÷
′
′
è uA u 1 u A u 2 ø
2
2
Thus:
−1 −1
2
G | b, u, y ~ IW ((S + V ) , v + q) (16.26)
2 u u u
which again is in the form of a p-dimensional inverted Wishart distribution with v + q
u
2
−1
degrees of freedom and scale parameter (S + V ).
u u
16.3.3 Numerical illustration
Example 16.2
Using the data in Example 5.1 and the variance components, the application of
Gibbs sampling to estimating variance components and predicting breeding values
is illustrated. Uniform priors are assumed for the variance components such that
v = v = −3 and V = V = 0. A flat prior is assumed for b, and u is assumed to be
e u e u
normally distributed.
Processing data and accumulating right-hand side (rhs) and diagonals (Diag) for
level j of sex of calf effects as:
Use of Gibbs Sampling 269