Assessing variable activity for Bayesian regression trees

TL;DR

This paper introduces an efficient, interpretable method for assessing variable importance in Bayesian Additive Regression Trees using Sobol' indices, addressing limitations of traditional one-way counts.

Contribution

It derives analytic Sobol' index expressions for BART, linking them to one-way counts and proposing a new ranking method for variable importance.

Findings

Analytic Sobol' indices for BART are computationally feasible.

The new ranking method preserves Sobol'-based importance order.

Comparison shows improved variable importance assessment over traditional counts.

Abstract

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.