# Trees, forests, and impurity-based variable importance

arXiv: https://arxiv.org/abs/2001.04295 · HTML: https://mlsanity.com/paper/2001.04295/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jan 2020.

## Built on by (most cited first, 16 of 16)

- [MDA for random forests: inconsistency, and a practical solution via the Sobol-MDA](https://mlsanity.com/paper/2102.13347/) (2021)
- [ControlBurn: Feature Selection by Sparse Forests](https://mlsanity.com/paper/2107.00219/) (2021)
- [A cautionary tale on fitting decision trees to data from additive models: generalization lower bounds](https://mlsanity.com/paper/2110.09626/) (2021)
- [Bridging Breiman's Brook: From Algorithmic Modeling to Statistical Learning](https://mlsanity.com/paper/2102.12328/) (2021)
- [From global to local MDI variable importances for random forests and when they are Shapley values](https://mlsanity.com/paper/2111.02218/) (2021)
- [Interpretable Models Capable of Handling Systematic Missingness in Imbalanced Classes and Heterogeneous Datasets](https://mlsanity.com/paper/2206.02056/) (2022)
- [Shapley Curves: A Smoothing Perspective](https://mlsanity.com/paper/2211.13289/) (2022)
- [MMD-based Variable Importance for Distributional Random Forest](https://mlsanity.com/paper/2310.12115/) (2023)
- [Topological Data Analysis for Anomaly Detection in Host-Based Logs](https://mlsanity.com/paper/2204.12919/) (2022)
- [FACT: High-Dimensional Random Forests Inference](https://mlsanity.com/paper/2207.01678/) (2022)
- [Stacked Autoencoder Based Multi-Omics Data Integration for Cancer Survival Prediction](https://mlsanity.com/paper/2207.04878/) (2022)
- [Subgroup analysis methods for time-to-event outcomes in heterogeneous randomized controlled trials](https://mlsanity.com/paper/2401.11842/) (2024)
- [Aggregate Models, Not Explanations: Improving Feature Importance Estimation](https://mlsanity.com/paper/2602.11760/) (2026)
- [Evolutionary Optimization of High-Coverage Budgeted Classifiers](https://mlsanity.com/paper/2110.13067/) (2021)
- [Dynamic Features Adaptation in Networking: Toward Flexible training and Explainable inference](https://mlsanity.com/paper/2510.08303/) (2025)
- [Just add noise: Debiasing tree-based variable importance in mixed data](https://mlsanity.com/paper/2609.14083/) (2026)

## Abstract (arXiv)

Tree ensemble methods such as random forests [Breiman, 2001] are very popular to handle high-dimensional tabular data sets, notably because of their good predictive accuracy. However, when machine learning is used for decision-making problems, settling for the best predictive procedures may not be reasonable since enlightened decisions require an in-depth comprehension of the algorithm prediction process. Unfortunately, random forests are not intrinsically interpretable since their prediction results from averaging several hundreds of decision trees. A classic approach to gain knowledge on this so-called black-box algorithm is to compute variable importances, that are employed to assess the predictive impact of each input variable. Variable importances are then used to rank or select variables and thus play a great role in data analysis. Nevertheless, there is no justification to use random forest variable importances in such way: we do not even know what these quantities estimate. In this paper, we analyze one of the two well-known random forest variable importances, the Mean Decrease Impurity (MDI). We prove that if input variables are independent and in absence of interactions, MDI provides a variance decomposition of the output, where the contribution of each variable is clearly identified. We also study models exhibiting dependence between input variables or interaction, for which the variable importance is intrinsically ill-defined. Our analysis shows that there may exist some benefits to use a forest compared to a single tree.