Bayesian Additive Regression Trees (BART) is a popular Bayesian non-parametric regression model that is commonly used in causal inference and beyond. Its strong predictive performance is supported by theoretical guarantees that its posterior distribution concentrates around the true regression function at optimal rates under various data generative settings and for appropriate prior choices. In this paper, we show that the BART sampler often converges slowly, confirming empirical observations by other researchers. Assuming discrete covariates, we show that, while the BART posterior concentrates on a set comprising all optimal tree structures (smallest bias and complexity), the Markov chain's hitting time for this set increases with $n$ (training sample size), under several common data generative settings. As $n$ increases, the approximate BART posterior thus becomes increasingly different from the exact posterior (for the same number of MCMC samples), contrasting with earlier concentration results on the exact posterior. This contrast is highlighted by our simulations showing worsening frequentist undercoverage for approximate posterior intervals and a growing ratio between the MSE of the approximate posterior and that obtainable by artificially improving convergence via averaging multiple sampler chains. Finally, based on our theoretical insights, possibilities are discussed to improve the BART sampler convergence performance.

翻译：贝叶斯加性回归树（BART）是一种广泛应用于因果推断及其他领域的贝叶斯非参数回归模型。其优异的预测性能得到了理论保证的支持：在各种数据生成设定及适当的先验选择下，其后验分布能够以最优速率集中于真实回归函数周围。本文证明，BART采样器往往收敛缓慢，这证实了其他研究者的实证观察。在假设协变量离散的前提下，我们证明：虽然BART后验集中于包含所有最优树结构（最小偏差与复杂度）的集合，但在几种常见数据生成设定下，马尔可夫链对该集合的命中时间随$n$（训练样本量）增加而增长。随着$n$增大，近似BART后验因而与精确后验（在相同MCMC样本量下）的差异逐渐扩大，这与先前关于精确后验的集中性结论形成对比。这种反差通过我们的模拟实验得以凸显：近似后验区间的频率主义覆盖不足逐渐恶化，且近似后验的均方误差与通过人工平均多个采样链以改善收敛所能获得的均方误差之比持续增长。最后，基于我们的理论见解，本文探讨了提升BART采样器收敛性能的潜在途径。