Bayesian Additive Regression Trees (BART) is a powerful statistical model that leverages the strengths of Bayesian inference and regression trees. It has received significant attention for capturing complex non-linear relationships and interactions among predictors. However, the accuracy of BART often comes at the cost of interpretability. To address this limitation, we propose ANOVA Bayesian Additive Regression Trees (ANOVA-BART), a novel extension of BART based on the functional ANOVA decomposition, which is used to decompose the variability of a function into different interactions, each representing the contribution of a different set of covariates or factors. Our proposed ANOVA-BART enhances interpretability, preserves and extends the theoretical guarantees of BART, and achieves comparable prediction performance. Specifically, we establish that the posterior concentration rate of ANOVA-BART is nearly minimax optimal, and further provides the same convergence rates for each interaction that are not available for BART. Moreover, comprehensive experiments confirm that ANOVA-BART is comparable to BART in both accuracy and uncertainty quantification, while also demonstrating its effectiveness in component selection. These results suggest that ANOVA-BART offers a compelling alternative to BART by balancing predictive accuracy, interpretability, and theoretical consistency.

翻译：贝叶斯加性回归树（BART）是一种结合贝叶斯推断与回归树优势的强大统计模型，因其能够捕捉预测变量间复杂的非线性关系与交互作用而备受关注。然而，BART的准确性往往以可解释性为代价。为克服这一局限，我们提出基于函数方差分析分解的ANOVA贝叶斯加性回归树（ANOVA-BART）——该新型扩展模型通过函数方差分析将函数变异性分解为不同交互项，每项代表不同协变量或因子集合的贡献。我们提出的ANOVA-BART在提升可解释性的同时，保持并拓展了BART的理论保证，且获得了相当的预测性能。具体而言，我们证明ANOVA-BART的后验集中率近乎达到极小极大最优，并进一步为每个交互项提供了BART所不具备的相同收敛速率。此外，综合实验证实ANOVA-BART在预测精度与不确定性量化方面与BART相当，同时展现出优异的成分选择能力。这些结果表明，ANOVA-BART通过平衡预测准确性、可解释性与理论一致性，为BART提供了具有竞争力的替代方案。