Semiparametric Bayesian networks (SPBNs) integrate parametric and non-parametric probabilistic models, offering flexibility in learning complex data distributions from samples. In particular, kernel density estimators (KDEs) are employed for the non-parametric component. Under the assumption of data normality, the normal rule is used to determine the bandwidth matrix for KDEs in SPBNs. This matrix is the critical hyperparameter that controls the trade-off between bias and variance. However, real-world data often deviates from normality, potentially leading to suboptimal density estimation and reduced predictive performance. This paper presents the theoretical framework for applying state-of-the-art bandwidth selectors to SPBNs and evaluates their impact on the performance. We explore cross-validation and plug-in selectors approaches, assessing their effectiveness in enhancing the learning capability and applicability of SPBNs. To support this investigation, we have extended the open-source package PyBNesian for SPBNs with additional bandwidth selection techniques and conducted extensive experimental analyses. Our results demonstrate that the proposed bandwidth selectors leverage larger sample sizes more effectively than the normal rule, which, despite its robustness, plateaus with more data. In particular, unbiased cross-validation generally outperforms the normal rule, highlighting its advantage in large sample size scenarios. • We establish guarantees for SPBN density estimation using plug-in and CV bandwidths. • Bandwidth selection boosts SPBN parameter/structure learning beyond normal rule. • Proposed bandwidths improve with sample size, unlike the stagnating normal rule. • Despite stagnating performance, normal rule is robust/effective in structure learning. • Unbiased CV balances cost/performance; best overall, especially for large samples.
Alejandre et al. (Fri,) studied this question.