Gibbs sampling, adaptive rejection sampling and robustness to prior specification for a mixed linear model

Markov chain Monte-Carlo methods are increasingly being applied to make inferences about the marginal posterior distributions of parameters in quantitative genetic models. This paper considers the application of one such method, Gibbs sampling, to Bayesian inferences about parameters in a normal mixed linear model when a restriction is imposed on the relative values of the variance components. Two prior distributions are proposed which incorporate this restriction. For one of them, the adaptive rejection sampling technique is used to sample from the conditional posterior distribution of a variance ratio. Simulated data from a balanced sire model are used to compare different implementations of the Gibbs sampler and also inferences based on the two prior distributions. Marginal posterior distributions of the parameters are illustrated. Numerical results suggest that it is not necessary to discard any iterates, and that similar inferences are made using the two prior specifications.