Page 210 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

(BayesB). Other variations of the Bayesian methods such as BayesC and BayesCπ
(where some SNPs are assumed to have zero effects, and others are assumed to follow
a normal distribution) have been published by Habier et al. (2011). This section
presents some of these methods.

11.8.1 BayesA

Instead of the assumption of a normal distribution for SNP effects as in the SNP-
BLUP model, another possible assumption is that the distribution follows a Student’s
t-distribution. This allows for a higher probability of moderate to large SNP effects
than a normal distribution. However, the t-distribution is not easy to incorporate
into prediction of marker effects, so a mathematically tractable way of achieving this
2
is to assume that each SNP effect comes from a normal distribution but s can be
g
2
2
varied among the SNPs. Thus if s is large then gˆ will be large and if s is small,
g g
then gˆ will likely be small as it will regress towards zero (Hayes and Daetwyler,
2013). This leads to modelling the data at two levels: first at the level of the data
that is similar to SNP-BLUP to estimate the SNP effects and second at the variances
of the chromosome segments or SNPs, which are assumed to be different at every
segment or locus. The procedure uses a Gibbs sampling approach, which involves
sampling from the posterior distributions conditioned on other effects. If the reader
is not familiar with Gibbs sampling, they may want to read Chapter 16, where appli-
cation of the Gibbs sampling for the estimation of genetic parameters is discussed.
Thus given the linear model in Eqn 11.6, the conditional distribution that gener-
ates the data, y, is:
2
2
y | b,g,s ~ N(Xb + Zg + Rs )
e e
Prior distributions

Specification of the Bayesian model involves defining the prior distributions. Usually,
an improper or ‘flat’ prior distribution is assigned to b. Thus P(b) ~ constant.
The overall mean effect (b) is then sampled from the following conditional distri-
bution as:
2
2
2
X′Xb|g, s , s ,y ~ N(X′(y - Zg), X′Xs )
gi e e
Therefore:
ˆ
2 2 ( ,( ′ − 1 2
b g|,s gi ,s e y , ~ N b X X) s e ) (11.20)
ˆ
where b = (X′X) X′(y − Zg)
−1
−2
A scaled inverted chi distribution, χ (v, S) is usually used as priors for the vari-
ance components, with v being the degrees of freedom and S the scaled parameter
(Wang et al., 1993). Thus for the residual variance, prior uniform distribution (χ (−2, 0))
−2
or flat prior can be assumed. Sampling is then from the following conditional pos-
terior distribution:
2
s | e ~ χ − 2 n ( − 2, ′ i i (11.21)
e e )
i
e
194 Chapter 11