Semi-Linear Gaussian Bayesian Networks: Enhancing Flexibility in Node Dependencies via penalized B-Splines

Aitor Oviedo-Madrid; Mabel Morales-Otero; Jose Gonzalez-Gomariz; Ruben Armañanzas

cs.AIOct 22, 2026

Semi-Linear Gaussian Bayesian Networks: Enhancing Flexibility in Node Dependencies via penalized B-Splines

Authors: Aitor Oviedo-Madrid, Mabel Morales-Otero, Jose Gonzalez-Gomariz, Ruben Armañanzas

Organizations: Institute of Data Science and Artiﬁcial Intelligence (DATAI), University of Navarra, Pamplona, Spain · TECNUN School of Engineering, University of Navarra, San Sebastián, Spain

Abstract

Bayesian networks for continuous variables often rely on restrictive assumptions: linear Gaussian models lack flexibility to capture nonlinear dependencies, while non-parametric approaches provide greater flexibility at a high computational cost. This creates a gap for modeling scenarios where variables are approximately Gaussian but exhibit nonlinear relationships. We introduce Semi-Linear Gaussian Bayesian Networks (SLGBN), a novel class of continuous Bayesian networks designed to bridge this gap, preserving Gaussian conditional distributions while introducing nonlinear conditional mean functions. The proposed framework provides a unified probabilistic framework combining linear Gaussian and nonlinear Gaussian conditional models within the same Bayesian network Nonlinear conditional mean functions are estimated using generalized additive models (GAMs) with penalized B-splines. We present parameter estimation procedures for both types of nodes, as well as a greedy hill-climbing algorithm for learning the network structure under the SLGBN framework. To assess its performance, we compare SLGBN with other continuous Bayesian network models, including semiparametric Bayesian networks (SPBN) and Linear Gaussian Bayesian networks (LGBN). Experimental results evaluate both goodness of fit and the ability to recover the underlying network structure. The results show that, in the presence of nonlinear Gaussian relationships, SLGBN achieves better fit and improved structure recovery compared to alternative models, particularly when combined with the Extended Bayesian Information Criterion (EBIC*) score. Furthermore, by retaining Gaussian conditional distributions, SLGBN offers computational advantages over fully semiparametric approaches.

Figures & tables

Figure 1: Greedy Hill-Climbing: The figure shows four steps of the Greedy Hill-Climbing Algorithm and changes in the estimation of $X_{2}$ . Yellow nodes represent nonlinear nodes and gray nodes represent linear nodes. Changes in arcs are represented with red arrows and changes in types are represented by red contours. Step 1 starts with a linear $X_{2}$ node depending on $X_{1}$ . In step 2 node type is changed from linear to nonlinear and $X_{2}$ is estimated using B-splines. In step 3 arc ( $X_{1}$ , $X_{2}$ ) is reversed and $X_{2}$ loses its only parent. Its density is at this point estimated with sample mean but the node type is still nonlinear. Note that at this point type_change operator is not available for $X_{2}$ as it has no parents. In step 4 $X_{2}$ gains a new parent after adding ( $X_{3}$ , $X_{2}$ ) edge and is again nonlinearly estimated with B-splines, but now depending on $X_{3}$ .

Network	Number of nodes	Number of edges	Source
B5	5	5	Own elaboration
B8	8	12	Own elaboration
B_Insu	27	52	bnlearn

Table 1: Synthetic Networks.

Figure 2: B5 network. First synthetic network structure and node probability distributions.

Figure 3: B8 network. Second synthetic network structure and node probability distribution.

Figure 4: Fixed metrics. Evolution of the metrics in structure learning with fixed structure for different model-score configurations. Values represent the mean across all random seeds for each sample size. Rows correspond to metrics: a) Log-likelihood, b) THD (Lower is better). Columns correspond to networks: B5 (left), B8 (center), and B_insu (right). X axis shows sample sizes. Y axis represents metric values, being common for THD.

Figure 5: Full metrics. Evolution of metrics in full structure learning for different model-score configurations. Values represent the mean across all random seeds for each sample size. Rows correspond to metrics: a) Log-likelihood, b) THD (Lower is better), c) SHD (Lower is better), d) Precision, and e) Recall. Columns correspond to networks: B5 (left), B8 (center), and B_insu (right). X axis shows sample sizes. Y axis represents metric values, with common axis by network for all metrics except log-likelihood.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: BN_insu network. The structure of the insurance network from bnlean . Third synthetic network.

Node	Probability distribution	Node Type
Age	$\mathcal{N}(0,1)$	Linear
SocioEcon	$\mathcal{N}(0.5\cdot{\sin{(Age)}}+0.3\cdot{Age^{2}},1)$	Nonlinear
RiskAversion	$\mathcal{N}(\tanh{(Age)}+0.5\cdot{SocioEcon},1)$	Nonlinear
MakeModel	$\mathcal{N}(3+0.5\cdot{RiskAversion}+0.6\cdot{SocioEcon},1)$	Linear
Mileage	$\mathcal{N}(0,1)$	Linear
HomeBase	$\mathcal{N}(0.3\cdot{RiskAversion}+0.5\cdot{SocioEcon},1)$	Linear

Appendix

Table 2: Parametrization of B_insu.

Model-Score	Sample size	Ground LL	LL	THD
LG-BIC	200	-3462.53	-4339.37 $\pm$ 61.99	2.00 $\pm$ 0.00
LG-BIC	1000	-3462.53	-4277.49 $\pm$ 9.75	2.00 $\pm$ 0.00
LG-BIC	5000	-3462.53	-4266.94 $\pm$ 2.73	2.00 $\pm$ 0.00
LG-BIC	10000	-3462.53	-4265.84 $\pm$ 2.58	2.00 $\pm$ 0.00
LG-EBIC*	200	-3462.53	-4339.37 $\pm$ 61.99	2.00 $\pm$ 0.00
LG-EBIC*	1000	-3462.53	-4277.49 $\pm$ 9.75	2.00 $\pm$ 0.00

Appendix

Table 3: B5 Fixed.

Model-Score	Sample size	Ground LL	LL	THD
LG-BIC	200	-5515.31	-7188.73 $\pm$ 176.45	3.00 $\pm$ 0.00
LG-BIC	1000	-5515.31	-7032.95 $\pm$ 12.06	3.00 $\pm$ 0.00
LG-BIC	5000	-5515.31	-7018.39 $\pm$ 8.14	3.00 $\pm$ 0.00
LG-BIC	10000	-5515.31	-7016.94 $\pm$ 5.17	3.00 $\pm$ 0.00
LG-EBIC*	200	-5515.31	-7188.73 $\pm$ 176.45	3.00 $\pm$ 0.00
LG-EBIC*	1000	-5515.31	-7032.95 $\pm$ 12.06	3.00 $\pm$ 0.00

Appendix

Table 4: B8 Fixed.

Model-Score	Sample size	Ground LL	LL	THD
LG-BIC	200	-38246.71	-40247.69 $\pm$ 58.36	11.00 $\pm$ 0.00
LG-BIC	1000	-38246.71	-40009.88 $\pm$ 17.00	11.00 $\pm$ 0.00
LG-BIC	5000	-38246.71	-39965.91 $\pm$ 4.11	11.00 $\pm$ 0.00
LG-BIC	10000	-38246.71	-39961.14 $\pm$ 3.62	11.00 $\pm$ 0.00
LG-EBIC*	200	-38246.71	-40247.69 $\pm$ 58.36	11.00 $\pm$ 0.00
LG-EBIC*	1000	-38246.71	-40009.88 $\pm$ 17.00	11.00 $\pm$ 0.00

Appendix

Table 5: B_insu Fixed.

Model-Score	Sample size	Ground LL	LL	THD	SHD	Precision	Recall
LG-BIC	200	-3462.53	-4344.83 $\pm$ 63.74	2.00 $\pm$ 0.00	0.95 $\pm$ 0.76	0.96 $\pm$ 0.10	0.86 $\pm$ 0.09
LG-BIC	1000	-3462.53	-4278.18 $\pm$ 9.59	2.00 $\pm$ 0.00	1.00 $\pm$ 0.46	0.97 $\pm$ 0.07	0.83 $\pm$ 0.07
LG-BIC	5000	-3462.53	-4267.58 $\pm$ 3.54	2.00 $\pm$ 0.00	1.05 $\pm$ 0.39	0.96 $\pm$ 0.08	0.83 $\pm$ 0.07
LG-BIC	10000	-3462.53	-4266.25 $\pm$ 2.73	2.00 $\pm$ 0.00	1.05 $\pm$ 0.39	0.98 $\pm$ 0.06	0.81 $\pm$ 0.04
LG-EBIC*	200	-3462.53	-4337.12 $\pm$ 54.29	2.00 $\pm$ 0.00	0.85 $\pm$ 0.49	0.99 $\pm$ 0.04	0.84 $\pm$ 0.08
LG-EBIC*	1000	-3462.53	-4277.08 $\pm$ 9.43	2.00 $\pm$ 0.00	0.90 $\pm$ 0.31	1.00 $\pm$ 0.00	0.82 $\pm$ 0.06

Appendix

Table 6: B5 Free.

Model-Score	Sample size	Ground LL	LL	THD	SHD	Precision	Recall
LG-BIC	200	-5515.31	-7206.62 $\pm$ 158.62	3.00 $\pm$ 0.00	5.50 $\pm$ 1.82	0.82 $\pm$ 0.12	0.71 $\pm$ 0.09
LG-BIC	1000	-5515.31	-7032.67 $\pm$ 13.20	3.00 $\pm$ 0.00	6.15 $\pm$ 2.03	0.75 $\pm$ 0.10	0.77 $\pm$ 0.06
LG-BIC	5000	-5515.31	-7007.75 $\pm$ 10.85	3.00 $\pm$ 0.00	7.85 $\pm$ 1.79	0.64 $\pm$ 0.06	0.82 $\pm$ 0.06
LG-BIC	10000	-5515.31	-7003.69 $\pm$ 7.45	3.00 $\pm$ 0.00	8.00 $\pm$ 1.84	0.62 $\pm$ 0.06	0.87 $\pm$ 0.05
LG-EBIC*	200	-5515.31	-7201.46 $\pm$ 165.02	3.00 $\pm$ 0.00	4.40 $\pm$ 1.39	0.94 $\pm$ 0.10	0.68 $\pm$ 0.06
LG-EBIC*	1000	-5515.31	-7038.44 $\pm$ 13.25	3.00 $\pm$ 0.00	4.70 $\pm$ 1.69	0.87 $\pm$ 0.11	0.73 $\pm$ 0.06

Appendix

Table 7: B8 Free.

Model-Score	Sample size	Ground LL	LL	THD	SHD	Precision	Recall
LG-BIC	200	-38246.71	-40433.97 $\pm$ 130.31	11.00 $\pm$ 0.00	34.55 $\pm$ 7.79	0.65 $\pm$ 0.08	0.74 $\pm$ 0.04
LG-BIC	1000	-38246.71	-39909.71 $\pm$ 29.50	11.00 $\pm$ 0.00	37.35 $\pm$ 7.11	0.61 $\pm$ 0.07	0.81 $\pm$ 0.02
LG-BIC	5000	-38246.71	-39817.52 $\pm$ 30.12	11.00 $\pm$ 0.00	46.50 $\pm$ 11.14	0.54 $\pm$ 0.08	0.86 $\pm$ 0.02
LG-BIC	10000	-38246.71	-39792.27 $\pm$ 27.43	11.00 $\pm$ 0.00	44.55 $\pm$ 12.21	0.55 $\pm$ 0.08	0.88 $\pm$ 0.02
LG-EBIC*	200	-38246.71	-40471.27 $\pm$ 143.57	11.00 $\pm$ 0.00	24.40 $\pm$ 3.49	0.84 $\pm$ 0.05	0.65 $\pm$ 0.03
LG-EBIC*	1000	-38246.71	-39937.79 $\pm$ 44.53	11.00 $\pm$ 0.00	27.55 $\pm$ 5.39	0.73 $\pm$ 0.07	0.77 $\pm$ 0.02

Appendix

Table 8: B_insu Free.

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Dirichlet Process Mixtures within Semiparametric Bayesian Networks: A Novel Conditional Density Estimation Approach

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Computational Intelligence Group Departamento de Inteligencia Artificial Universidad Politécnica de Madrid Campus de Montegancedo, Boadilla del Monte 28660 Madrid, Spain

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Semiparametric Bayesian networks (SPBNs) are flexible probabilistic graphical models for representing joint distributions of continuous variables through lower-dimensional conditional probability distributions (CPDs). Existing SPBN models use kernel density estimators (KDEs) for non-parametric CPD modeling, but their one-component-per-observation mixture representation can lead to increased inference costs and density estimation errors, which can be limiting. In this work, we define a new CPD modeling approach for non-parametric components based on Bayesian non-parametric Dirichlet Process Mixtures (DPMs) within SPBNs. DPMs provide a data-adaptive clustering of observations, yielding a more compact representation while retaining the flexibility of non-parametric density estimation, unlike KDE. This results in lower inference costs while improving density estimation performance. We implement the proposed DPM-based SPBN (DPM-SPBN) by extending the open-source PyBNesian package and evaluate it on synthetic and real-world data. For a given network structure, DPM-based CPDs significantly outperform state-of-the-art Bayesian network models for continuous variables in density estimation, achieving lower absolute log-likelihood errors on synthetic data. We further investigate the impact of the proposed CPD modeling on structure learning methods relying on CPD-based scores. DPM-based SPBNs achieve lower structure learning and density estimation errors than KDE-based SPBNs. Experiments on real-world data corroborate these findings, identifying DPM-based CPD modeling as the most effective alternative among the evaluated non-parametric approaches for continuous variables.

Abstract

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