如果你用朴素贝叶斯处理高基数分类数据，HEB-NB能自动调平滑参数，比固定Laplace更准，论文还给了理论保证和实测数据。
HEB-NB通过Type-II最大似然学习Dirichlet先验浓度，实现数据自适应的平滑，解决了Laplace等固定平滑在高基数数据上的偏差问题。理论证明其非渐近误差界匹配经验分布极小极大率，并给出有限样本风险级严格优于Laplace的证明。在31个UCI和OpenML基准上，HEB-NB取得最佳平均Friedman排名，高基数数据集对数损失降低达22.1%。HEB-AODE扩展也一致优于原始AODE。结合互信息加权，top-1 ECE降低41%-70%。
Hierarchical Empirical-Bayes Naive Bayes: Minimax Smoothing and Calibration with AODE Extension
The Naive Bayes (NB) classifier remains a standard choice for categorical data, yet its widely used smoothing rules, such as Laplace, Lidstone, Krichevsky-Trofimov, and the $m$-estimate, all prescribe a fixed smoothing strength that ignores feature cardinality, sample size, and class imbalance, inducing a non-vanishing bias on modern high-cardinality tabular data. We propose hierarchical empirical-Bayes Naive Bayes (HEB-NB), in which each class-feature conditional probability is smoothed by a Dirichlet prior whose concentration is learned data-adaptively via Type-II maximum likelihood, enabling principled information sharing across classes while retaining closed-form inference. We further introduce HEB average one-dependence estimators (HEB-AODE), showing that the adaptive smoothing transfers cleanly to structural relaxations of NB. Theoretically, we establish a non-asymptotic $\ell_1$ error bound for HEB-NB matching the empirical-distribution minimax rate plus a vanishing data-adaptive bias, together with a matching Laplace-tight lower bound that yields a finite-sample, risk-level strict separation from Laplace. We further derive a plug-in excess Bayes-risk bound via total-variation tensorization and a population top-1 expected calibration error (ECE) corollary. Empirically, across 31 UCI and OpenML benchmarks, HEB-NB attains the best average Friedman rank on probabilistic metrics, with up to 22.1% log-loss reductions on high-cardinality datasets and consistent improvements of HEB-AODE over vanilla AODE. Combining HEB-NB with mutual-information weighting reduces top-1 ECE by 41%-70%, demonstrating substantial gains in probabilistic accuracy and calibration.