多元变系数BART与图形马蹄先验
Multivariate Varying-Coefficient BART with Graphical Horseshoe Priors
AI总结：
提出多变量变系数BART框架，用独立BART集成建模非线性系数曲面，图形马蹄先验捕捉残差稀疏依赖，并证明后验收缩率。
摘要：
现代多元回归问题涉及多个相关结果，其回归效应不仅是非线性、异质性和结果特异性的，而且结果之间的残差依赖具有科学意义。现有的多元贝叶斯树方法通常只解决部分问题：有些方法在结果之间强加树架构的实质性共享，当响应依赖于不同的预测变量或效应修饰因子时，这过于严格；而其他方法则适应残差依赖但保留更简单的均值结构。本文开发了multiVCBART，一个多元变系数贝叶斯加性回归树框架，联合建模灵活的结果特异性系数曲面和稀疏残差精度矩阵。系数矩阵$B(x)$的每个条目由独立的BART集成表示，允许预测效应随修饰因子$x$在不同结果间非线性变化，而精度矩阵$\Omega$上的图形马蹄先验捕捉简约的残差条件依赖。为了允许高效计算，我们引入了一个采样器，将多元高斯似然简化为一系列标量伪响应更新，将树回拟合与图形马蹄步骤解耦。理论上，我们建立了具有联合估计残差依赖的多元BART模型的第一个后验收缩率，证明了对底层平滑性和结构稀疏性的近极小极大适应性。经验上，multiVCBART在稀疏高维数据集上优于现有的多元树模型和贝叶斯SUR竞争者。最后，在癌症药物敏感性基因组学数据集的再分析中，我们的方法识别了不同的生物标志物信号并恢复了一个连贯的残差药理学网络。
英文摘要：
Modern multivariate regression problems involve several related outcomes whose regression effects are not only nonlinear, heterogeneous, and outcome-specific, but also where the residual dependence among outcomes is scientifically meaningful. Existing multivariate Bayesian tree-based methods typically address only part of this problem: some impose substantial sharing of tree architecture across outcomes, which is overly restrictive when responses depend on distinct predictors or effect modifiers, while others accommodate residual dependence but retain simpler mean structures. This paper develops multiVCBART, a multivariate varying-coefficient Bayesian additive regression tree framework that jointly models flexible outcome-specific coefficient surfaces and a sparse residual precision matrix. Each entry of the coefficient matrix $B(x)$ is represented by an independent BART ensemble, allowing predictor effects to vary nonlinearly with modifiers $x$ across outcomes, while a Graphical Horseshoe prior on the precision matrix $Ω$ captures parsimonious residual conditional dependence. To permit efficient computation, we introduce a sampler that reduces the multivariate Gaussian likelihood to a sequence of scalar pseudo-response updates, decoupling the tree backfitting from the Graphical Horseshoe step. Theoretically, we establish the first posterior contraction rates for a multivariate BART model with jointly estimated residual dependence, proving near-minimax adaptation to underlying smoothness and structural sparsity. Empirically, multiVCBART outperforms existing multivariate tree models and Bayesian SUR competitors on sparse, high-dimensional datasets. Finally, in a re-analysis of the Genomics of Drug Sensitivity in Cancer dataset, our method identifies distinct biomarker signals and recovers a coherent residual pharmacologic network.
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