title: EXACT MINIMAX ESTIMATION OF THE PREDICTIVE DENSITY IN SPARSE GAUSSIAN MODELS.
abstract: We consider estimating the predictive density under Kullback-Leibler loss in anℓ0sparse 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.
authors.affiliation: Department of Data Sciences and Operations, Marshall School of Business, University of Southern California, Los Angeles, California 90089-0809, USA.
authors.forename: Gourab
authors.initials: G
authors.lastname: Mukherjee
authors.affiliation: Department of Statistics, Sequoia Hall, 390 Serra Mall, Stanford University, Stanford, California 94305-4065, USA.
authors.forename: Iain M
authors.initials: IM
authors.lastname: Johnstone
keywords: Predictive density; high-dimensional; minimax; mutual information; plug-in risk; risk diversification; sparsity; thresholding