  We propose a Bayesian approach using improper priors for hierarchical linear
mixed models with flexible random effects and residual error distributions. The
error distribution is modelled using scale mixtures of normals, which can
capture tails heavier than those of the normal distribution. This
generalisation is useful to produce models that are robust to the presence of
outliers. The case of asymmetric residual errors is also studied. We present
general results for the propriety of the posterior that also cover cases with
censored observations, allowing for the use of these models in the contexts of
popular longitudinal and survival analyses. We consider the use of copulas with
flexible marginals for modelling the dependence between the random effects, but
our results cover the use of any random effects distribution. Thus, our paper
provides a formal justification for Bayesian inference in a very wide class of
models (covering virtually all of the literature) under attractive prior
structures that limit the amount of required user elicitation.
