XMSE-Aware Adaptive Empirical Bayes Estimation

2026-06-26 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Engineering & Applied Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, extended

Summary

An XMSE-aware mixed estimator is introduced, designed to adaptively combine Maximum Likelihood (ML) and Empirical Bayes (EB) shrinkage. It addresses the issue where kernel-based EB estimation can perform worse than ML due to kernel-parameter misalignment, a problem identified by Excess Mean Squared Error (XMSE) analysis. The proposed estimator's fixed-weight XMSE is a scalar quadratic, allowing for a closed-form oracle mixing weight that guarantees performance no worse than both ML and the base EB estimator at the XMSE scale. A plug-in implementation, utilizing finite-sample XMSE approximations, demonstrates consistency and achieves a second-order oracle regret rate. Experimental results on FIR system identification, including Silverbox and Cascaded Tanks benchmarks, confirm the estimator's ability to utilize regularization when beneficial and revert to ML when kernel misspecification occurs, although a finite-sample calibration failure mode was observed.

Key takeaway

For Machine Learning Engineers or Research Scientists deploying regularized estimators in system identification, you should consider this XMSE-aware mixed estimator. It provides a robust safeguard, allowing you to benefit from Empirical Bayes regularization when kernels are well-aligned, while gracefully retreating to Maximum Likelihood under kernel misspecification. This approach mitigates the risk of performance degradation from poorly chosen regularization, offering a principled way to adaptively balance bias and variance. Be mindful of finite-sample calibration, which may require trace correction.

Key insights

An XMSE-aware mixed estimator adaptively blends ML and EB shrinkage, using second-order risk to mitigate kernel misspecification.

Principles

• XMSE analysis informs adaptive estimator design.
• Interpolate base estimators for robustness.
• Restrict mixing weights to prevent risk inflation.

Method

Compute ML and base EB estimates, then plug-in XMSE components. Use these to derive a projected mixing weight for the combined estimator.

In practice

• Protect against kernel mismatch in system ID.
• Apply trace correction for plug-in bias.
• Use XMSE criterion for kernel selection.

Topics

• Empirical Bayes
• Maximum Likelihood Estimation
• Excess Mean Squared Error
• System Identification
• Kernel-based Regularization
• Adaptive Estimators

Best for: AI Scientist, Research Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.