提出自适应贝叶斯估计器,在核不匹配时自动退回到最大似然法,提升稳健性。
XMSE-Aware Adaptive Empirical Bayes Estimation

- 设计混合估计器,根据核对齐度动态权衡最大似然与贝叶斯收缩
- 理论证明其均方误差风险优于传统方法,且在有限样本下一致收敛
- 适用于信号处理等场景,尤其适合核函数可能错配的实证研究
经验贝叶斯(EB)估计器在第一阶渐近风险上可媲美最大似然(ML),但在第二阶表现差异显著:近期的超额均方误差(XMSE)分析表明,当核函数与真实参数不对齐时,基于核的EB估计可能劣于ML。本文将此诊断转化为设计原则,提出一种感知XMSE的混合估计器,可在ML与EB收缩之间插值。其固定权重下的XMSE为标量二次型,给出闭式最优混合权重,该权重在XMSE尺度上不劣于ML和基线EB估计。基于有限样本XMSE近似的插件实现被证明是一致的,且具有内部最优权重的二阶最优后悔率。进一步建立后悔界向选定权重处固定权重风险曲线的转移,并扩展至紧致核族及有限与增长核字典,给出高概率最优边界。通过有限脉冲响应模拟,对比SURE调参、硬选择与迹校正基线,结合公开的Silverbox与Cascaded Tanks基准测试,结果表明该估计器在正则化有效时保留大部分优势,而在核误设时退回到ML,有限样本分析已验证其性能。
原文摘要 · Abstract (English)
Empirical Bayes (EB) estimators can match the first-order asymptotic risk of maximum likelihood (ML) while behaving very differently at second order: recent excess mean squared error (XMSE) analysis shows that kernel-based EB estimation may be worse than ML when the kernel is poorly aligned with the true parameter. This paper turns that diagnostic into a design principle. We propose an XMSE-aware mixed estimator that interpolates between ML and EB shrinkage. Its fixed-weight XMSE is a scalar quadratic, yielding a closed-form oracle mixing weight that is no worse than both ML and the base EB estimator at the XMSE scale. A plug-in implementation based on finite-sample XMSE approximations is proved consistent, with a second-order oracle regret rate for an interior oracle weight. We further establish a transfer of the regret bound to the fixed-weight risk curve evaluated at the selected weight, a thresholded boundary rule, and extensions to compact kernel families and to finite and growing kernel dictionaries with high-probability oracle bounds. Finite impulse response simulations with SURE-tuned, hard-selection, and trace-corrected baselines, together with the public Silverbox and Cascaded Tanks benchmarks, show that the proposed estimator retains most of the benefit of regularization when it is helpful and retreats toward ML under kernel misspecification, with an identified finite-de analyzed on the benchmarks.
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