Page 276 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

16Use of Gibbs Sampling

in Variance Component

Estimation and Breeding

Value Prediction

16.1 Introduction

Gibbs sampling is a numerical integration method and is one of several Markov chain
Monte Carlo (MCMC) methods. They involve drawing samples from specified dis-
tributions; hence they are called Monte Carlo and are referred to as Markov chain
because each sample depends on the previous sample. Specifically, Gibbs sampling
involves generating random drawings from marginal posterior distributions through
iteratively sampling from the conditional posterior distributions. For instance, given
that Q′ = (Q , Q ) and P(Q , Q ) is the joint distribution of Q and Q , Gibbs sampling
1 2 1 2 1 2
involves sampling from the full conditional posterior distributions of Q , P(Q |Q ) and
1 1 2
Q , P(Q |Q ).
2 2 1
Thus given that the joint posterior distribution is known to proportionality, the
conditional distributions can be generated. However, defining the joint density
involves the use of Bayes’ thereom. In general, given that the probability of two events
occurring together, P(B, Y), is:

P(B,Y) = P(B)P(Y|B) = P(Y)P(B|Y)
then:
P(B|Y) = P(B)P(Y|B)/P(Y) (16.1)
Equation 16.1 implies that inference about the variable B depends on the prior prob-
ability of its occurrence, P(B). Given that observations on Y are available, this prior
probability is then updated to obtain the posterior probability or density of B, (P(B|Y).
Equation 16.1 is commonly expressed as:
P(B|Y) ∝ P(B)P(Y|B) (16.2)
as the denominator is not a function of B. Therefore, the posterior density of B is
proportional to the prior probability of B times the conditional distribution of Y
given B. Assuming that B in Eqn 16.2 is replaced by W, a vector of parameters, such
that W′ = (W ,W ,W ), and that the joint posterior distribution is known to propor-
1 2 3
tionality (Eqn 16.2), the full conditional probabilities needed for the Gibbs sampler
can be generated for each parameter as P(W |W ,W ,Y), P(W |W ,W ,Y) and
1 2 3 2 1 3
[0]
[0]
[0]
P(W |W ,W ,Y). Assuming starting values W , W and W , the implementation
3 1 2 1 2 3
of the Gibbs sampler involves iterating the following loop:
260 © R.A. Mrode 2014. Linear Models for the Prediction of Animal Breeding Values,
3rd Edition (R.A. Mrode)