  We consider estimating the predictive density under Kullback-Leibler loss in
a high-dimensional Gaussian model. Decision theoretic properties of the
within-family prediction error -- the minimal risk among estimates in the class
$\mathcal{G}$ of all Gaussian densities are discussed. We show that in sparse
models, the class $\mathcal{G}$ is minimax sub-optimal. We produce
asymptotically sharp upper and lower bounds on the within-family prediction
errors for various subfamilies of $\mathcal{G}$. Under mild regularity
conditions, in the sub-family where the covariance structure is represented by
a single data dependent parameter $\Shat=\dhat \cdot I$, the Kullback-Leiber
risk has a tractable decomposition which can be subsequently minimized to yield
optimally flattened predictive density estimates. The optimal predictive risk
can be explicitly expressed in terms of the corresponding mean square error of
the location estimate, and so, the role of shrinkage in the predictive regime
can be determined based on point estimation theory results. Our results
demonstrate that some of the decision theoretic parallels between predictive
density estimation and point estimation regimes can be explained by second
moment based concentration properties of the quadratic loss.
