MARGINAL INFERENCES ABOUT VARIANCE-COMPONENTS IN A MIXED LINEAR-MODELUSING GIBBS SAMPLING

Arguing from a Bayesian viewpoint, Gianola and Foulley (1990) derived a new method for estimation of variance components in a mixed linear m odel: variance estimation from integrated likelihoods (VEIL). Inferenc e is based on the marginal posterior distribution of each of the varia nce components. Exact analysis requires numerical integration. In this paper, the Gibbs sampler, a numerical procedure for generating margin al distributions from conditional distributions, is employed to obtain marginal inferences about variance components in a general univariate mixed linear model. All needed conditional posterior distributions ar e derived. Examples based on simulated data sets containing varying am ounts of information are presented for a one-way sire model. Estimates of the marginal densities of the variance components and of functions thereof are obtained, and the corresponding distributions are plotted . Numerical results with a balanced sire model suggest that convergenc e to the marginal posterior distributions is achieved with a Gibbs seq uence length of 20, and that Gibbs sample sizes ranging from 300 - 3 0 00 may be needed to appropriately characterize the marginal distributi ons.