揭示贝叶斯加性回归树的高斯过程极限，实现可解析计算的替代方案。
On the Gaussian process limit of Bayesian Additive Regression Trees
- 推导出BART先验协方差函数，首次实现无限树极限下的精确高斯过程表示。
- 极限模型在固定配置下性能弱于标准BART，但调参后可达到相当水平。
- 提供解析似然，简化建模且避免复杂MCMC采样，适合需高效推断的场景。
贝叶斯加性回归树（BART）是一种日益流行的非参数贝叶斯回归方法，本质是决策树之和，可视为提升的贝叶斯版本。当树的数量趋于无穷时，其等价于高斯过程（GP）回归。该极限虽已知，却未带来实用分析或应用。本文首次推导并计算了BART先验协方差函数，据此将无限树极限下的BART实现为GP回归。实验证明，在固定配置下该极限性能劣于标准BART，但通过自然的GP方式调参后，其表现可与BART竞争。使用GP作为BART代理的优势在于解析似然，显著简化模型构建，并绕过复杂的BART MCMC算法。本研究为理解与改进BART及GP回归开辟新路径。代码实现已开源于Python包lsqfitgp。
原文摘要 · Abstract (English)
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.
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