# MMD-based Variable Importance for Distributional Random Forest

arXiv: https://arxiv.org/abs/2310.12115 · HTML: https://mlsanity.com/paper/2310.12115/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 58 references that are in the map.
- First posted on arXiv 18 Oct 2023.

## Builds on (most cited first, 14 of 14)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Estimation and Inference of Heterogeneous Treatment Effects using Random Forests](https://mlsanity.com/paper/1510.04342/) (2015)
- [Generalized Random Forests](https://mlsanity.com/paper/1610.01271/) (2016)
- [Panning for Gold: Model-X Knockoffs for High-dimensional Controlled Variable Selection](https://mlsanity.com/paper/1610.02351/) (2016)
- [Unrestricted Permutation forces Extrapolation: Variable Importance Requires at least One More Model, or There Is No Free Variable Importance](https://mlsanity.com/paper/1905.03151/) (2019)
- [Consistency of random forests](https://mlsanity.com/paper/1405.2881/) (2014)
- [A general framework for inference on algorithm-agnostic variable importance](https://mlsanity.com/paper/2004.03683/) (2020)
- [Correlation and variable importance in random forests](https://mlsanity.com/paper/1310.5726/) (2013)
- [Distributional Random Forests: Heterogeneity Adjustment and Multivariate Distributional Regression](https://mlsanity.com/paper/2005.14458/) (2020)
- [Metrizing Weak Convergence with Maximum Mean Discrepancies](https://mlsanity.com/paper/2006.09268/) (2020)
- [SHAFF: Fast and consistent SHApley eFfect estimates via random Forests](https://mlsanity.com/paper/2105.11724/) (2021)
- [Trees, forests, and impurity-based variable importance](https://mlsanity.com/paper/2001.04295/) (2020)
- [Confidence and Uncertainty Assessment for Distributional Random Forests](https://mlsanity.com/paper/2302.05761/) (2023)

## Built on by (most cited first, 2 of 2)

- [Distributional Random Forests for Complex Survey Designs](https://mlsanity.com/paper/2512.08179/) (2025)
- [Kernel of Partition Paths: A Unified Representation for Tree Ensembles](https://mlsanity.com/paper/2606.18853/) (2026)

## Abstract (arXiv)

Distributional Random Forest (DRF) is a flexible forest-based method to estimate the full conditional distribution of a multivariate output of interest given input variables. In this article, we introduce a variable importance algorithm for DRFs, based on the well-established drop and relearn principle and MMD distance. While traditional importance measures only detect variables with an influence on the output mean, our algorithm detects variables impacting the output distribution more generally. We show that the introduced importance measure is consistent, exhibits high empirical performance on both real and simulated data, and outperforms competitors. In particular, our algorithm is highly efficient to select variables through recursive feature elimination, and can therefore provide small sets of variables to build accurate estimates of conditional output distributions.