Exact minimax estimation of the predictive density in sparse Gaussian models

TL;DR

This paper derives the exact minimax risk and optimal predictive density estimators for sparse Gaussian models under Kullback-Leibler loss, revealing new decision-theoretic phenomena and strategies that outperform plug-in methods.

Contribution

It provides explicit formulas for the minimax risk, asymptotically least favorable priors, and constructs optimal predictive densities outside the Gaussian family.

Findings

Explicit first-order minimax risk and constants derived.

Optimal strategies involve threshold-based predictive densities.

Plug-in estimates are suboptimal for predictive density estimation.

Abstract

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.