Let y represent an n x 1 observable random vector that follows the mixed linear model y = X[beta] + Zs + e. Here, X and Z are specified matrices, [beta] is a column vector of unknown fixed parameters, and s and e are statistically independent multivariate normal random column vectors with E(e) = 0, E(s) = 0, var(e) = [sigma][subscript]sp e2I, and var(s) = [sigma][subscript]sp e2[gamma]A, where A is a known, positive definite matrix. Further, [sigma][subscript]sp e2 and [gamma] are unknown, scalar-valued parameters such that [sigma][subscript]sp e2> 0 and 0 ≤ [gamma] ≤ u, where u is a specified constant;The problem considered is that of Bayesian inference for a predictable random variable w = [lambda][superscript]'[beta] + [phi][superscript]'s, for the case where the prior distribution of [beta] is non-informative. It is shown that the problem of computing the posterior mean, variance, and density of w can be reduced to that of numerically evaluating one-dimensional integrals, provided the distribution of [sigma][subscript]sp e2 and [gamma] is of the general form G[subscript]1([gamma])([sigma][subscript]sp e2)[superscript] G2([gamma])exp-(2[sigma][subscript]sp e2)[superscript]-1G[subscript]3([gamma]). Here, G[subscript]1([gamma]), G[subscript]2([gamma]) and G[subscript]3([gamma]) are functions of [gamma];The mean of the posterior distribution of w can be approximated by w = [lambda][superscript]'[beta] + [phi][superscript]' s, where [beta] and s are the mode of the joint posterior distribution of [beta] and s. If there is no upper bound on [gamma], the computation of w does not require numerical integration. The problem of computing a 100(1 - [alpha])% highest posterior density (HPD) credible set for w can be reduced to that of solving a constrained minimization problem. An algorithm is described for obtaining 100(1 - [alpha])% HPD credible sets;The feasibility of using the Bayesian methodology was evaluated by applying the methodology to predict the genetic merit of dairy bulls. A secondary objective in the analysis of these data was to compare the classical and Bayesian approaches. Classical and Bayesian predictions, along with prediction error variances and posterior variances for the breeding values of 1,028 Holstein bulls were obtained for two traits: milk production and number of days open. For these data, the Bayesian and the classical predictions were very similar.