The posterior distribution of the fixed and random effects in a mixed-effects linear model

The subject of this paper is Bayesian inference about the fixed and random effects of a mixed-effects linear statistical model with two variance components. It is assumed that a priori the fixed effects have a noninformative distribution and that the reciprocals of the variance components are distributed independently (of each other and of the fixed effects) as gamma random variables. It is shown that techniques similar to those employed in a ridge analysis of a response surface can be used to construct a one-dimensional curve that contains all of the stationary points of the posterior density of the random effects. The "ridge analysis" (of the posterior density) can be useful (from a computational standpoint) in finding the number and the locations of the stationary points and can be very informative about various features of the posterior density. Depending on what is revealed by the ridge analysis, a multivariate normal or multivariate-t distribution that is centered at a posterior mode may provide a satisfactory approximation to the posterior distribution of the random effects (which is of the poly-t form).