  We consider estimating the predictive density under Kullback-Leibler loss in
an $\ell_0$ sparse Gaussian sequence model. Explicit expressions of the first
order minimax risk along with its exact constant, asymptotically least
favorable priors and optimal predictive density estimates are derived. Compared
to the sparse recovery results involving point estimation of the normal mean,
new decision theoretic phenomena are seen. Suboptimal performance of the class
of plug-in density estimates reflects the predictive nature of the problem and
optimal strategies need diversification of the future risk. We find that
minimax optimal strategies lie outside the Gaussian family but can be
constructed with threshold predictive density estimates. Novel minimax
techniques involving simultaneous calibration of the sparsity adjustment and
the risk diversification mechanisms are used to design optimal predictive
density estimates.
