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.

通过将随机效应引入贝叶斯网络学习，该研究提出了一种新的方法，可以增强结构学习并改进模型规范，从而应用于复杂的农学数据集结构，提高决策支持工具的价值。

利用混合效应模型和分层聚类学习具有异质农学数据集的贝叶斯网络

Learning Bayesian Networks with Heterogeneous Agronomic Data Sets via  Mixed-Effect Models and Hierarchical Clustering

In regression analysis of counts, a lack of simple and efficient algorithms
for posterior computation has made Bayesian approaches appear unattractive and
thus underdeveloped. We propose a lognormal and gamma mixed negative binomial
(NB) regression model for counts, and present efficient closed-form Bayesian
inference; unlike conventional Poisson models, the proposed approach has two
free parameters to include two different kinds of random effects, and allows
the incorporation of prior information, such as sparsity in the regression
coefficients. By placing a gamma distribution prior on the NB dispersion
parameter r, and connecting a lognormal distribution prior with the logit of
the NB probability parameter p, efficient Gibbs sampling and variational Bayes
inference are both developed. The closed-form updates are obtained by
exploiting conditional conjugacy via both a compound Poisson representation and
a Polya-Gamma distribution based data augmentation approach. The proposed
Bayesian inference can be implemented routinely, while being easily
generalizable to more complex settings involving multivariate dependence
structures. The algorithms are illustrated using real examples.

本研究提出了一种基于负二项式拟合的 lognormal 和 gamma 混合模型，应用了贝叶斯推断，实现了回归系数的稀疏性先验等效果，提高了贝叶斯方法计算的简洁性和效率。