  We study frequentist risk properties of predictive density estimators for
mean mixtures of multivariate normal distributions, involving an unknown
location parameter $\theta \in \mathbb{R}^d$, and which include multivariate
skew normal distributions. We provide explicit representations for Bayesian
posterior and predictive densities, including the benchmark minimum risk
equivariant (MRE) density, which is minimax and generalized Bayes with respect
to an improper uniform density for $\theta$. For four dimensions or more, we
obtain Bayesian densities that improve uniformly on the MRE density under
Kullback-Leibler loss. We also provide plug-in type improvements, investigate
implications for certain type of parametric restrictions on $\theta$, and
illustrate and comment the findings based on numerical evaluations.
