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arXiv · 2512.13992

Bayesian Global-Local Regularization

Jyotishka Datta·Nick Polson·Vadim Sokolov
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Abstract

We propose a unified framework for global-local regularization that bridges the gap between classical techniques -- such as ridge regression and the nonnegative garotte -- and modern Bayesian hierarchical modeling. By estimating local regularization strengths via marginal likelihood under order constraints, our approach generalizes Stein's positive-part estimator and provides a principled mechanism for adaptive shrinkage in high-dimensional settings. We establish that this isotonic empirical Bayes estimator achieves near-minimax risk (up to logarithmic factors) over sparse ordered model classes, constituting a significant advance in high-dimensional statistical inference. Applications to orthogonal polynomial regression demonstrate the methodology's flexibility, while our theoretical results clarify the connections between empirical Bayes, shape-constrained estimation, and degrees-of-freedom adjustments.

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Jyotishka Datta, Nick Polson, Vadim Sokolov. 2025-12-16. Bayesian Global-Local Regularization. https://arxiv.org/abs/2512.13992

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