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Local MDI+: Local Feature Importances for Tree-Based Models

This paper introduces Local MDI+ (LMDI+), a novel method that extends the global MDI+ framework to provide stable, structure-aware local feature importances for tree-based models, outperforming existing baselines in identifying instance-specific predictive features and enabling advanced interpretability use cases like counterfactual generation.

Original authors: Zhongyuan Liang, Zachary T. Rewolinski, Abhineet Agarwal, Tiffany M. Tang, Bin Yu

Published 2026-05-28

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CC BY 4.0

Original authors: Zhongyuan Liang, Zachary T. Rewolinski, Abhineet Agarwal, Tiffany M. Tang, Bin Yu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a very smart, complex robot (a "tree-based model" like a Random Forest) that makes predictions about things like house prices or medical diagnoses. You know the robot is good at guessing, but it's a bit of a "black box." You want to know: "Why did the robot make this specific guess for this specific person?"

This is where Local Feature Importance (LFI) comes in. It's like asking the robot to point to the specific ingredients in a recipe that made the cake taste sweet.

The Problem with Current Methods

Currently, the most popular ways to ask the robot "why?" are like LIME and TreeSHAP. The paper argues these methods are a bit like a clumsy detective:

They guess by shaking things: They try to explain a prediction by randomly messing up the data (perturbations) and seeing how the robot reacts. It's like trying to figure out why a car engine is loud by randomly hitting it with a hammer and listening to the noise.
They are unstable: If you ask the detective the same question twice, they might give you two different answers because their "random shaking" was slightly different.
They miss the real signal: Sometimes they blame the wrong ingredients (noise) instead of the real ones (signal).

The New Solution: Local MDI+

The authors propose a new method called Local MDI+. Think of this as a master architect who doesn't just guess; they look at the actual blueprints of the robot.

Here is how it works, using a simple analogy:

The Blueprint (The Tree Structure): Instead of guessing, Local MDI+ looks inside the robot's brain. It sees the decision trees (the flowcharts the robot uses to make decisions).
The Translation (Linear Regression): The paper discovered a clever trick: you can translate the robot's complex decision tree into a simple math equation (a linear model). Imagine taking a complex, winding maze and realizing it's actually just a straight line if you look at it from the right angle.
The "Out-of-Bag" Safety Net: The robot is trained on a bunch of data. Usually, it gets tested on the same data it learned from, which is like a student taking a test on the exact questions they memorized. Local MDI+ uses a special "out-of-bag" technique, where it tests the robot on data it didn't memorize. This prevents the robot from cheating and gives a truer score of what actually matters.
The Result: By combining the robot's internal map with this "safety net" math, Local MDI+ can tell you exactly which features mattered for this specific person, without needing to randomly shake the data.

Why is this better? (The Paper's Claims)

The paper ran this new method against the old ones (LIME, TreeSHAP, and Local MDI) on 12 real-world datasets (like housing data and medical data). Here is what they found:

It finds the real clues: When the researchers tested the methods on fake data where they knew the answer, Local MDI+ was much better at finding the true "signal" features and ignoring the "noise." It was like a detective who actually finds the fingerprint instead of guessing based on a shadow.
It's consistent: If you run the robot 100 times with different random seeds, Local MDI+ gives you the same ranking of important features every time. The old methods would change their minds. This is crucial for trust; you don't want a doctor's diagnosis changing just because the computer restarted.
It handles messy data: When features are highly correlated (like "height in inches" and "height in centimeters"), old methods get confused and blame the wrong one. Local MDI+ stays calm and ranks the important ones correctly.
It works for "What-If" questions: The paper showed that Local MDI+ is better at finding "counterfactuals." This means if a loan was rejected, Local MDI+ can tell you the smallest, most realistic change needed to get approved (e.g., "increase income by $500" rather than "change your name"). It finds the path of least resistance.
It finds hidden groups: In a case study with Miami housing data, the method grouped houses into neighborhoods that actually made sense geographically and financially. It found that houses in one cluster were cheap because they were small and far away, while another cluster was expensive because they were huge, even if they were in similar areas.

The Bottom Line

Local MDI+ is a new tool for explaining how tree-based AI models make decisions. Instead of guessing by randomly breaking things, it looks at the model's internal structure, uses a smart mathematical translation, and checks its work on data it hasn't seen before. The result is an explanation that is more accurate, more stable, and more trustworthy than the current popular methods.

Note: The paper explicitly mentions the implementation is available in the imodels package and works with Random Forests and Gradient Boosting models. It does not claim to work on deep learning models or specific clinical outcomes beyond the general examples provided.

Technical Summary: Local MDI+ (LMDI+)

Problem Statement

Tree-based ensembles, such as Random Forests (RFs) and Gradient Boosting, remain the state-of-the-art for tabular data due to their predictive performance and computational efficiency. Consequently, they are widely deployed in high-stakes domains like healthcare and criminal justice, necessitating trustworthy local interpretations.

Current Local Feature Importance (LFI) methods, such as LIME and TreeSHAP, face significant limitations:

Reliance on Perturbations: These model-agnostic approaches rely on random perturbations or feature permutations to approximate model behavior. This ignores the model's internal structure, leading to unstable explanations and failure to identify true signal features in the underlying Data Generating Processes (DGPs).
Instability: Existing methods often produce inconsistent feature rankings across repeated model fits with different random seeds.
Bias in Tree-Based Methods: Traditional tree-based local importance (Local MDI) suffers from biases where features with high entropy or low correlation receive inflated importance scores. Furthermore, tree-based models are inefficient at capturing additive and smooth functions.

While the global feature importance method MDI+ addressed these biases by exploiting an equivalence between decision trees and least squares on a transformed node basis, it remains a global metric. It cannot explain sample-specific outcomes, which often depend on unique individual characteristics.

Methodology: The LMDI+ Framework

The authors propose Local MDI+ (LMDI+), a novel extension of the MDI+ framework to the sample-specific setting. Developed under the Predictability-Computability-Stability (PCS) framework for veridical data science, LMDI+ aims to identify instance-specific signal features faithfully and consistently without relying on perturbations.

The framework leverages the equivalence between decision tree predictions and Ordinary Least Squares (OLS) regression on a transformed basis (stump functions). The LMDI+ procedure involves three key steps:

Expanded Representation: For each tree in an ensemble, the method constructs a feature map $\tilde{\Psi}(X)$ $\tilde{Ψ} (X)$ that includes:
- The transformed stump functions $\Psi(X; S)$ derived from the tree structure (based on in-bag and Out-of-Bag (OOB) samples).
- The raw feature values $x_k$ appended to the map. This augmentation allows the model to capture linear and additive structures that pure tree splits might miss.
Regularized GLM Fitting: Instead of using OLS, LMDI+ fits a regularized Generalized Linear Model (GLM) (e.g., ElasticNet) on the transformed data. This step mitigates overfitting and handles correlated features better than standard tree-based importance measures.
Sample-Specific Attribution: For a specific observation $x$ , the importance of the $k$ -th feature is calculated as the contribution of that feature to the prediction within the linearized model:
$\text{LMDI}^+_k(x) = \tilde{\Psi}^{(k)}(x)^\top \hat{\beta}^{(k)}_\lambda$
where $\hat{\beta}^{(k)}_\lambda$ represents the GLM coefficients. If a feature never appears in a split, its contribution is zero. The final ensemble score is the average of these scores across all trees.

Key Contributions and Results

The paper validates LMDI+ across twelve real-world benchmark datasets (six regression, six classification) and various synthetic settings.

1. Superior Identification of Signal Features

Synthetic Experiments: In data-inspired simulations with varying signal-to-noise ratios (SNR), sample sizes, and response functions (linear, polynomial, and locally-spiky-sparse), LMDI+ consistently achieved higher AUROC scores in distinguishing signal from non-signal features compared to LIME, TreeSHAP, and Local MDI.
Robustness to Correlation: In settings with highly correlated features, LMDI+ correctly ranked signal features as most important. In contrast, baselines like LIME and TreeSHAP often incorrectly prioritized uncorrelated non-signal features when correlation strength was high ( $\rho \ge 0.9$ ).
Real-World Performance: Using a "remove-and-retrain" evaluation technique, LMDI+ demonstrated that features it selected were more predictive of individual outcomes. When retraining models on masked data retaining only the top $k\%$ of features, LMDI+ yielded higher $R^2$ (regression) and AUROC (classification) scores than baselines, with an average 10% improvement in predictive performance when using only selected features.

2. Enhanced Stability

LMDI+ produced the most consistent feature rankings across repeated model fits with different random seeds.
Stability experiments showed that LMDI+ selected the fewest unique features across different model fits, indicating that its importance rankings are robust to modeling variations, a critical requirement for trustworthy interpretations.

3. Generalizability and Ablation

Component Analysis: Ablation studies confirmed that each component of LMDI+ (OOB samples, raw feature augmentation, and regularized GLMs) contributes to performance gains.
Model Agnosticism within Trees: The improvements extended beyond Random Forests to Gradient Boosting models, demonstrating the framework's applicability to various tree ensembles.

4. Practical Applications

Counterfactual Explanations: By prioritizing signal features, LMDI+ identified counterfactuals (minimal changes to flip a prediction) that required smaller modifications (lower $\ell_1$ distance) compared to baselines.
Subgroup Discovery: In a case study on the Miami Housing dataset, clustering samples based on LMDI+ scores revealed homogeneous subgroups. Fitting simple OLS models on these subgroups significantly reduced prediction error (MSE) compared to a global model, with LMDI+ outperforming other LFI methods in subgroup homogeneity.

Significance and Claims

The paper claims that LMDI+ addresses the critical gap between global tree-based importance and the need for local, sample-specific interpretability. By moving away from perturbation-based approximations and instead exploiting the internal linear structure of tree ensembles, LMDI+ offers:

Faithfulness: It more accurately identifies the true signal features driving individual predictions.
Stability: It provides consistent explanations across model runs, reducing the risk of unreliable decision-making in high-stakes settings.
Actionability: It enables practical use cases such as generating efficient counterfactuals and discovering homogeneous data subgroups.

The authors position LMDI+ as a step toward veridical data science, ensuring that local interpretations are not only computable but also predictable and stable. While the method introduces a one-time overhead for fitting the GLM and transforming the basis, the inference phase for new samples is efficient (simple linear multiplication), making it viable for practical deployment where the model structure remains fixed.