EXACT MINIMAX ESTIMATION OF THE PREDICTIVE DENSITY IN SPARSE GAUSSIAN MODELS

Area of Science:

• Statistics
• Machine Learning
• Information Theory

Background:

• Estimating predictive density is crucial for probabilistic forecasting.
• Sparse Gaussian sequence models are widely used in signal processing and econometrics.
• Kullback-Leibler loss is a standard metric for evaluating probability distributions.

Purpose of the Study:

• To derive minimax risk expressions for predictive density estimation in ℓ0 sparse Gaussian sequence models.
• To identify optimal predictive density estimates and understand their properties.
• To compare predictive density estimation with traditional sparse recovery methods.

Main Methods:

• Derivation of explicit expressions for first-order minimax risk.
• Identification of asymptotically least favorable priors.
• Construction of optimal predictive density estimates using thresholding techniques.

Main Results:

• New decision-theoretic phenomena emerge compared to point estimation.
• Plug-in density estimates show suboptimal performance due to the predictive nature of the problem.
• Minimax optimal strategies require diversification of future risk and lie outside the Gaussian family.

Conclusions:

• Optimal predictive density estimation necessitates strategies beyond standard Gaussian models.
• Threshold predictive density estimates are key to achieving minimax optimality.
• Novel minimax techniques are developed for simultaneous calibration of sparsity and risk diversification.