Sobol' indices and Shapley effects are attractive methods of assessing how a function depends on its various inputs. The existing literature contains various estimators for these two classes of sensitivity indices, but few estimators of Sobol' indices and no estimators of Shapley effects are computationally tractable for moderate-to-large input dimensions. This article provides a Shapley-effect estimator that is computationally tractable for a moderate-to-large input dimension. The estimator uses a metamodel-based approach by first fitting a Bayesian Additive Regression Trees model which is then used to compute Shapley-effect estimates. This article also establishes posterior contraction rates on a large function class for this Shapley-effect estimator and for the analogous existing Sobol'-index estimator. Finally, this paper explores the performance of these Shapley-effect estimators on four different test functions for moderate-to-large input dimensions and number of observations.

翻译：Sobol'指标和Shapley效应是评估函数如何依赖其各输入变量的有效方法。现有文献包含多种针对这两类敏感性指标的估计量，但对于中等至大输入维度，仅有少数Sobol'指标估计量（且尚无Shapley效应估计量）具备计算可行性。本文提出一种在中等至大输入维度下计算可行的Shapley效应估计量。该估计量基于元模型方法：首先拟合贝叶斯加性回归树模型，进而计算Shapley效应估计值。本文还在一个广泛的函数类上为所提出的Shapley效应估计量及现有类似的Sobol'指标估计量建立了后验收缩率。最后，本文通过四个测试函数，在中等至大输入维度和观测样本数下，探究了该Shapley效应估计量的性能表现。