Title: Marginal inferences about variance components in a mixed linear model using Gibbs sampling
Abstract: Arguing from a Bayesian viewpoint, Gianola and Foulley (1990) derived a new method for estimation of variance components in a mixed linear model: variance estimation from integrated likelihoods (VEIL).Inference is based on the marginal posterior distri- bution of each of the variance components.Exact analysis requires numerical integration.In this paper, the Gibbs sampler, a numerical procedure for generating marginal distri- butions from conditional distributions, is employed to obtain marginal inferences about variance components in a general univariate mixed linear model.All needed conditional posterior distributions are derived.Examples based on simulated data sets containing varying amounts 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 convergence to the marginal posterior distributions is achieved with a Gibbs sequence length of 20, and that Gibbs sample sizes ranging from 300 -3 000 may be needed to appropriately characterize the marginal distributions.variance components / linear models / Bayesian methods / marginalization / Gibbs sampler R.ésumé -Inférences marginales sur des composantes de variance dans un modèle linéaire mixte à l'aide de l'échantillonnage de Gibbs.Partant d'un point de vue bayésien, Gianola et Foulley (1990) ont établi une nouvelle méthode d'estimation des composantes de variance dans un modèle linéaire mixte: estimation de variance par les vraisemblances intégrées (VEIL).L'inférence est basée sur la distribution marginale a posteriori de chacune des composantes de variance, ce qui oblige à des intégrations numériques pour arriver aux solutions exactes.Dans cet article, l'échantillonnage de Gibbs, qui est une procédure numérique pour générer des distributions marginales à partir de distributions