We propose some extensions to semi-parametric models based on Bayesian additive regression trees (BART). In the semi-parametric BART paradigm, the response variable is approximated by a linear predictor and a BART model, where the linear component is responsible for estimating the main effects and BART accounts for non-specified interactions and non-linearities. Previous semi-parametric models based on BART have assumed that the set of covariates in the linear predictor and the BART model are mutually exclusive in an attempt to avoid poor coverage properties and reduce bias in the estimates of the parameters in the linear predictor. The main novelty in our approach lies in the way we change the tree-generation moves in BART to deal with this bias and resolve non-identifiability issues between the parametric and non-parametric components, even when they have covariates in common. This allows us to model complex interactions involving the covariates of primary interest, both among themselves and with those in the BART component. Our novel method is developed with a view to analysing data from an international education assessment, where certain predictors of students' achievements in mathematics are of particular interpretational interest. Through additional simulation studies and another application to a well-known benchmark dataset, we also show competitive performance when compared to regression models, alternative formulations of semi-parametric BART, and other tree-based methods. The implementation of the proposed method is available at \url{https://github.com/ebprado/CSP-BART}.

翻译：本文提出了基于贝叶斯加性回归树（BART）的半参数模型扩展方法。在半参数BART框架中，响应变量通过线性预测项和BART模型共同近似，其中线性成分负责估计主效应，而BART则处理未指定的交互作用和非线性关系。以往基于BART的半参数模型通常假设线性预测项与BART模型中的协变量集合互不重叠，以避免线性预测项参数估计的覆盖性不足和偏差问题。本方法的核心创新在于通过改进BART的树生成机制来处理这种偏差，并解决参数分量与非参数分量之间的不可识别性问题——即使两者包含共同协变量时依然有效。这使得我们能够对关键协变量之间及其与BART成分协变量之间的复杂交互作用进行建模。新方法的开发着眼于国际教育评估数据分析需求，其中某些影响学生数学成绩的预测因子具有特殊的解释意义。通过额外的模拟研究以及在经典基准数据集上的应用验证，本方法在线性回归模型、其他半参数BART变体以及基于树模型的对比中均表现出竞争优势。方法实现代码已发布于 \url{https://github.com/ebprado/CSP-BART}。