  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.
