Rejoinder to “Nonparametric variable importance assessment using machine learning techniques”

IL highlight the relationship between our proposed measures and the Breiman-Cutler family of variable importance measures (BC-VIMs). Indeed, apart from how the reducedmodel is defined, our procedures do fall under the umbrella of BC-VIMs. However, we arrive at these procedures from a somewhat different perspective than in the existing literature, and this distinction has important repercussions on both interpretation and inference. While we make this point explicit and expand on it in various directions in WGSC, we summarize it below. For this, it is useful to delineate the various definitions and objectives of variable importance assessment—in this discussion, we focus on what we believe are the two predominant perspectives on variable importance. Traditionally, the objective in variable importance assessment is to quantify the importance of a variable (or set of variables) within the confines of a given (possibly black box) prediction algorithm. For example, BC-VIMs play a key role in helping to interpret a fitted algorithm, leading to their tremendous popularity in the machine learning literature. The BC-VIM estimand must be interpreted relative to the fitted algorithm, and helps us to understand the extent to which a fitted model makes use of features. In the notation of IL, a BC-VIM is typically given by