Estimating Shapley effects for moderate-to-large input dimensions

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.