Hierarchical Empirical-Bayes Naive Bayes: Minimax Smoothing and Calibration with AODE Extension

• URL: http://arxiv.org/abs/2608.11162v1
• Date: Tue, 11 Aug 2026 17:21:31 GMT
• Title: Hierarchical Empirical-Bayes Naive Bayes: Minimax Smoothing and Calibration with AODE Extension
• Abstract summary: We propose hierarchical empirical-Bayes Naive Bayes (HEB-NB)<n>Each class-feature conditional probability is smoothed by a Dirichlet prior whose concentration is learned data-adaptively.<n>Across 31 UCI and OpenML benchmarks, HEB-NB attains the best average Friedman rank on probabilistic metrics.
• Score: 2.423836984571605
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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