  Bayesian Additive Regression Trees (BART) are non-parametric models that can
capture complex exogenous variable effects. In any regression problem, it is
often of interest to learn which variables are most active. Variable activity
in BART is usually measured by counting the number of times a tree splits for
each variable. Such one-way counts have the advantage of fast computations.
Despite their convenience, one-way counts have several issues. They are
statistically unjustified, cannot distinguish between main effects and
interaction effects, and become inflated when measuring interaction effects. An
alternative method well-established in the literature is Sobol' indices, a
variance-based global sensitivity analysis technique. However, these indices
often require Monte Carlo integration, which can be computationally expensive.
This paper provides analytic expressions for Sobol' indices for BART posterior
samples. These expressions are easy to interpret and are computationally
feasible. Furthermore, we will show a fascinating connection between
first-order (main-effects) Sobol' indices and one-way counts. We also introduce
a novel ranking method, and use this to demonstrate that the proposed indices
preserve the Sobol'-based rank order of variable importance. Finally, we
compare these methods using analytic test functions and the En-ROADS climate
impacts simulator.
