We consider the problem of predictive density estimation for normal models under Kullback-Leibler loss (KL loss) when the parameter space is constrained to a convex set. More particularly, we assume that X ∼ N p (µ, v x I) is observed and that we wish to estimate the density of Y ∼ N p (µ, v y I) under KL loss when µ is restricted to the convex set C ⊂ R p . We show that the best unrestricted invariant predictive density estimatorp U is dominated by the Bayes estimatorp π C associated to the

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... form prior π C on C. We also study so called plug-in estimators, giving conditions under which domination of one estimator of the mean vector µ over another under the usual quadratic loss, translates into a domination result for certain corresponding plug-in density estimators under KL loss. Risk comparisons and domination results are also made for comparisons of plug-in estimators and Bayes predictive density estimators. Additionally, minimaxity and domination results are given for the cases where: (i) C is a cone, and (ii) C is a ball.