Accounting for shared covariates in semi-parametric Bayesian additive regression trees

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 bias and poor coverage properties. The main novelty in our approach lies in the way we change the tree-generation moves in BART to deal with bias/confounding 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. Through synthetic and real-world examples, we demonstrate that the performance of our novel semi-parametric BART is competitive 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 https://github.com/ebprado/CSP-BART.