Research involving diverse but related data sets, where associations between covariates and outcomes may vary, is prevalent in various fields including agronomic studies. In these scenarios, hierarchical models, also known as multilevel models, are frequently employed to assimilate information from different data sets while accommodating their distinct characteristics. However, their structure extend beyond simple heterogeneity, as variables often form complex networks of causal relationships. Bayesian networks (BNs) provide a powerful framework for modelling such relationships using directed acyclic graphs to illustrate the connections between variables. This study introduces a novel approach that integrates random effects into BN learning. Rooted in linear mixed-effects models, this approach is particularly well-suited for handling hierarchical data. Results from a real-world agronomic trial suggest that employing this approach enhances structural learning, leading to the discovery of new connections and the improvement of improved model specification. Furthermore, we observe a reduction in prediction errors from 28\% to 17\%. By extending the applicability of BNs to complex data set structures, this approach contributes to the effective utilisation of BNs for hierarchical agronomic data. This, in turn, enhances their value as decision-support tools in the field.

翻译：涵盖不同但相关数据集的科学研究在农业研究等多个领域普遍存在，其中协变量与结果之间的关联可能因数据集而异。在此类场景中，层次模型（又称多水平模型）常被用于整合来自不同数据集的信息，同时保留其异质性特征。然而，这些模型的结构不仅限于简单的异质性，变量之间往往形成复杂的因果网络关系。贝叶斯网络通过有向无环图刻画变量间关联，为建模此类关系提供了强大框架。本研究提出一种将随机效应融入贝叶斯网络学习的新方法。该方法植根于线性混合效应模型，特别适用于处理层次结构数据。基于真实农业试验的结果表明，采用该方法可增强结构学习效果：既能发现新的变量连接，又能改进模型规范的准确性。此外，观察显示预测误差从28%降至17%。通过将贝叶斯网络的适用性扩展至复杂数据结构，该方法为层次化农业数据有效利用贝叶斯网络提供了支持，进而提升其作为该领域决策支持工具的价值。