Bayesian Additive Regression Trees (BART) is a nonparametric Bayesian regression technique of rising fame. It is a sum-of-decision-trees model, and is in some sense the Bayesian version of boosting. In the limit of infinite trees, it becomes equivalent to Gaussian process (GP) regression. This limit is known but has not yet led to any useful analysis or application. For the first time, I derive and compute the exact BART prior covariance function. With it I implement the infinite trees limit of BART as GP regression. Through empirical tests, I show that this limit is worse than standard BART in a fixed configuration, but also that tuning its hyperparameters in the natural GP way makes it competitive with BART. The advantage of using a GP surrogate of BART is the analytical likelihood, which simplifies model building and sidesteps the complex BART MCMC algorithm. More generally, this study opens new ways to understand and develop BART and GP regression. The implementation of BART as GP is available in the Python package lsqfitgp.

翻译：贝叶斯加性回归树（BART）是一种日益受到关注的非参数贝叶斯回归技术。它是一种决策树求和模型，在某种意义上可视为提升方法的贝叶斯版本。在无限多树的极限情况下，该模型等价于高斯过程（GP）回归。这一极限关系虽已知晓，但尚未催生任何有效的分析或应用。本文首次推导并计算了精确的BART先验协方差函数。基于此函数，我将BART的无限树极限实现为GP回归。通过实证测试表明：在固定配置下，该极限模型性能逊于标准BART；但若采用GP框架的自然方式调整其超参数，则能达到与BART相当的水平。使用BART的GP替代模型的核心优势在于其解析似然函数，这简化了模型构建过程，并规避了复杂的BART马尔可夫链蒙特卡罗算法。更广泛而言，本研究为理解和改进BART与GP回归开辟了新途径。BART的GP实现已发布于Python软件包lsqfitgp中。