This paper introduces Shrinkage Bayesian Causal Forest with Instrumental Variable (SBCF‑IV), a statistical method for finding and estimating subgroups whose causal effect differs from the average when treatment assignment is imperfect. The focus is on the Complier Average Causal Effect (CACE) — the effect for people who take the treatment when encouraged by an instrument and do not take it when not encouraged. Because compliance status is never directly observed, the authors study the conditional CACE (ccace), which describes how the complier effect varies with observed characteristics.

The researchers build on an earlier procedure called BCF‑IV that pairs a flexible ensemble model with a simple decision tree to get both accurate estimates and easy‑to‑read subgroups. SBCF‑IV changes two parts of that pipeline. First, it replaces the base ensemble with a “shrinkage” Bayesian Additive Regression Trees model that uses a sparsity‑inducing Dirichlet prior. In plain terms, the prior encourages the ensemble to concentrate its attention on a small number of covariates that actually matter for effect variation, instead of spreading splits across many irrelevant variables. Second, SBCF‑IV feeds the ensemble’s posterior split frequencies into the downstream CART decision tree (classification and regression tree) as variable‑level costs. That steers the final, interpretable partition to split on the covariates the ensemble found important.

At a high level, the method works in an instrumental‑variable setting with a binary randomized instrument and a binary treatment. The ensemble estimates two conditional objects that identify the ccace from observed data: the conditional intention‑to‑treat effect and the conditional share of compliers. The ccace is then recovered as the ratio of those objects under standard instrument assumptions. The sparsity prior helps when the covariate vector is high dimensional and only a few features drive heterogeneity.