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We consider estimating the predictive density under Kullback-Leibler loss inan $\ell_0$ sparse Gaussian sequence model. Explicit expressions of the firstorder minimax risk along with its exact constant, asymptotically leastfavorable priors and optimal predictive density estimates are derived. Comparedto the sparse recovery results involving point estimation of the normal mean,new decision theoretic phenomena are seen. Suboptimal performance of the classof plug-in density estimates reflects the predictive nature of the problem andoptimal strategies need diversification of the future risk. We find thatminimax optimal strategies lie outside the Gaussian family but can beconstructed with threshold predictive density estimates. Novel minimaxtechniques involving simultaneous calibration of the sparsity adjustment andthe risk diversification mechanisms are used to design optimal predictivedensity estimates.

Exact minimax estimation of the predictive density in sparse Gaussian models