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 ℓ1 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.
Empirical Bayes (EB) performs simultaneous inference across many related latent variables. Classical EB assumes that the likelihood p(x | z) is tractable. In many scientific applications, however, the likelihood is available only through a simulator. This paper develops EB for such implicit likelihoods. We introduce simulation-based empirical Bayes (SBEB), which connects nonparametric EB to simulation-based inference (SBI). SBEB computes EB estimates without an explicit density by using the observed data, simulator samples, and an amortized inference network. SBEB iteratively refines the fitted EB prior toward the population prior. With several scientific simulators and real-world data, we demonstrate that SBEB improves accuracy over SBI with a fixed prior.
The Generalized Naive Bayes (GNB) model was introduced for discrete and categorical random variables as an extension of classic Naive Bayes. We now accommodate the GNB framework to continuous explanatory variables. A central result of the paper is that structure learning of the GNB depends only on the pair copulas of the bi-variate marginals. We proved that the GNB structure can be assigned to the basis of a matroid, therefore we give greedy algorithms for finding the optimal GNB structure on the training data, in sense of minimizing Kullback-Leibler divergence. Three cases are considered: joint Gaussian distribution, then a more flexible model where we suppose the dependence structure to be described by a Gaussian copula with arbitrary marginals, and an even more flexible case where the joint continuous probability distribution is arbitrary, i.e. copula and marginal distributions are arbitrary. A method for model reduction, based on the newly introduced concept of GNB forest is given. We close the paper by comparing the newly introduced GNB classification results to other classical "glass-box" algorithms on real datasets.
Bayesian Networks (BNs) are of interest from an explainable AI viewpoint, offering transparent probabilistic models for decision support. Baymex is a recently introduced multi-objective evolutionary algorithm for learning discretized BNs, enabling experts to trade-off different objectives of interest, such as likelihood, model complexity, and prior beliefs. While Baymex has been shown to outperform state-of-the-art BN learning approaches, Baymex still 1) requires a lot of computation time and 2) has only been evaluated on synthetic data. To improve scalability, we introduce a parallelization strategy as well as a mechanism that enables adaptively steering optimization toward networks that overfit less. We furthermore reconfigure Baymex to train a BN classifier through multi-objective optimization of cross-entropy loss and the BIC complexity term so as to evaluate its performance on real-world clinical classification tasks. Besides observing speedups up to over 54 times on a 16-core CPU, comparisons against clinically familiar baselines (decision trees, logistic regression, naive Bayes, and random forests) on two open-source (RADCURE and SUPPORT) and one in-house dataset, show that Baymex obtains statistically similar or better predictive performance while producing compact, clinically inspectable BNs. Importantly, Baymex finds multiple plausible BN classifiers that contain predictors consistent with established clinical factors.
Damy M. F. Ha, Thalea Schlender, Yvette M. van der Linden +2