Singular Value Shrinkage Priors for Bayesian Prediction

TL;DR

This paper introduces singular value shrinkage priors for Bayesian matrix estimation, which improve prediction accuracy especially for low-rank matrices, by generalizing the Stein prior and ensuring minimax properties.

Contribution

The authors develop superharmonic priors that favor smaller singular values, extending the Stein prior to matrix parameters, and demonstrate their minimaxity and dominance over uniform priors.

Findings

Bayesian estimators with these priors are minimax.

The priors outperform uniform priors in finite samples.

They are particularly effective for low-rank matrices.

Abstract

We develop singular value shrinkage priors for the mean matrix parameters in the matrix-variate normal model with known covariance matrices. Our priors are superharmonic and put more weight on matrices with smaller singular values. They are a natural generalization of the Stein prior. Bayes estimators and Bayesian predictive densities based on our priors are minimax and dominate those based on the uniform prior in finite samples. In particular, our priors work well when the true value of the parameter has low rank.