In particular, interpreting Random Forests (RFs) [2] and its variants [14, 28, 27, 29, 1, 12] has become an important area of research due to the wide ranging applications of RFs invarious scientific areas, such asgenome-wide association studies (GWAS)[7],gene expression microarray[13,23],andgeneregulatorynetworks[9].

We thank the reviewers for their valuable feedback that will significantly improve our paper. This is indeed a limitation of Theorem 1. The CHIP data included in our simulation studies shows that MDI-oob works in this setting. We plan to add this plot in our supplementary material. Reviewers 2 and 3: Give theoretical/empirical evidence that MDI-oob can "debias" MDI. Empirically, we compute the MDI-oob for the first simulation.

Tree ensembles such as Random Forests have achieved impressive empirical success across a wide variety of applications. To understand how these models make predictions, people routinely turn to feature importance measures calculated from tree ensembles. It has long been known that Mean Decrease Impurity (MDI), one of the most widely used measures of feature importance, incorrectly assigns high importance to noisy features, leading to systematic bias in feature selection. In this paper, we address the feature selection bias of MDI from both theoretical and methodological perspectives. Based on the original definition of MDI by Breiman et al. [3] for a single tree, we derive a tight non-asymptotic bound on the expected bias of MDI importance of noisy features, showing that deep trees have higher (expected) feature selection bias than shallow ones. However, it is not clear how to reduce the bias of MDI using its existing analytical expression. We derive a new analytical expression for MDI, and based on this new expression, we are able to propose a debiased MDI feature importance measure using out-of-bag samples, called MDI-oob. For both the simulated data and a genomic ChIP dataset, MDI-oob achieves state-of-the-art performance in feature selection from Random Forests for both deep and shallow trees.