# Relating the Partial Dependence Plot and Permutation Feature Importance to the Data Generating Process

arXiv: https://arxiv.org/abs/2109.01433 · HTML: https://mlsanity.com/paper/2109.01433/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 42 references that are in the map.
- First posted on arXiv 3 Sep 2021.

## Builds on (most cited first, 10 of 10)

- [All Models are Wrong, but Many are Useful: Learning a Variable's Importance by Studying an Entire Class of Prediction Models Simultaneously](https://mlsanity.com/paper/1801.01489/) (2018)
- [Model-Agnostic Interpretability of Machine Learning](https://mlsanity.com/paper/1606.05386/) (2016)
- [Visualizing the Effects of Predictor Variables in Black Box Supervised Learning Models](https://mlsanity.com/paper/1612.08468/) (2016)
- [Accounting for Variance in Machine Learning Benchmarks](https://mlsanity.com/paper/2103.03098/) (2021)
- [Cross-validation: what does it estimate and how well does it do it?](https://mlsanity.com/paper/2104.00673/) (2021)
- [Model-agnostic Feature Importance and Effects with Dependent Features -- A Conditional Subgroup Approach](https://mlsanity.com/paper/2006.04628/) (2020)
- [Testing Conditional Independence in Supervised Learning Algorithms](https://mlsanity.com/paper/1901.09917/) (2019)
- [Floodgate: inference for model-free variable importance](https://mlsanity.com/paper/2007.01283/) (2020)
- [Sampling, Intervention, Prediction, Aggregation: A Generalized Framework for Model-Agnostic Interpretations](https://mlsanity.com/paper/1904.03959/) (2019)
- [Nonparametric Feature Impact and Importance](https://mlsanity.com/paper/2006.04750/) (2020)

## Built on by (most cited first, 24 of 24)

- [Dear XAI Community, We Need to Talk! Fundamental Misconceptions in Current XAI Research](https://mlsanity.com/paper/2306.04292/) (2023)
- [Scientific Inference With Interpretable Machine Learning: Analyzing Models to Learn About Real-World Phenomena](https://mlsanity.com/paper/2206.05487/) (2022)
- [A Guide to Feature Importance Methods for Scientific Inference](https://mlsanity.com/paper/2404.12862/) (2024)
- [Performance is not enough: the story told by a Rashomon quartet](https://mlsanity.com/paper/2302.13356/) (2023)
- [RHALE: Robust and Heterogeneity-aware Accumulated Local Effects](https://mlsanity.com/paper/2309.11193/) (2023)
- [Ultra-marginal Feature Importance: Learning from Data with Causal Guarantees](https://mlsanity.com/paper/2204.09938/) (2022)
- [Confident Feature Ranking](https://mlsanity.com/paper/2307.15361/) (2023)
- [A Statistical Case Against Empirical Human-AI Alignment](https://mlsanity.com/paper/2502.14581/) (2025)
- [Why You Should Not Trust Interpretations in Machine Learning: Adversarial Attacks on Partial Dependence Plots](https://mlsanity.com/paper/2404.18702/) (2024)
- [Interpretable Machine Learning for Survival Analysis](https://mlsanity.com/paper/2403.10250/) (2024)
- [Fast Estimation of Partial Dependence Functions using Trees](https://mlsanity.com/paper/2410.13448/) (2024)
- [Explaining and visualizing black-box models through counterfactual paths](https://mlsanity.com/paper/2307.07764/) (2023)
- [On the Robustness of Global Feature Effect Explanations](https://mlsanity.com/paper/2406.09069/) (2024)
- [Analyzing Error Sources in Global Feature Effect Estimation](https://mlsanity.com/paper/2603.15057/) (2026)
- [xplainfi: Feature Importance and Statistical Inference for Machine Learning in R](https://mlsanity.com/paper/2603.15306/) (2026)
- [MinShap: A Shapley-Based Framework for Feature Redundancy](https://mlsanity.com/paper/2604.15107/) (2026)
- [Innovations in Agricultural Forecasting: A Multivariate Regression Study on Global Crop Yield Prediction](https://mlsanity.com/paper/2312.02254/) (2023)
- [mlr3summary: Concise and interpretable summaries for machine learning models](https://mlsanity.com/paper/2404.16899/) (2024)
- [Statistics and explainability: a fruitful alliance](https://mlsanity.com/paper/2404.19301/) (2024)
- [Targeted Learning for Variable Importance](https://mlsanity.com/paper/2411.02221/) (2024)
- [Comparing Model-agnostic Feature Selection Methods through Relative Efficiency](https://mlsanity.com/paper/2508.14268/) (2025)
- [Interval Prediction of Annual Average Daily Traffic on Local Roads via Quantile Random Forest with High-Dimensional Spatial Data](https://mlsanity.com/paper/2510.18548/) (2025)
- [Imputation Uncertainty in Interpretable Machine Learning Methods](https://mlsanity.com/paper/2512.17689/) (2025)
- [A Resilience-as-a-Service assessment framework for coordinated disruption response in interdependent urban transit systems](https://mlsanity.com/paper/2606.08849/) (2026)

## Abstract (arXiv)

Scientists and practitioners increasingly rely on machine learning to model data and draw conclusions. Compared to statistical modeling approaches, machine learning makes fewer explicit assumptions about data structures, such as linearity. However, their model parameters usually cannot be easily related to the data generating process. To learn about the modeled relationships, partial dependence (PD) plots and permutation feature importance (PFI) are often used as interpretation methods. However, PD and PFI lack a theory that relates them to the data generating process. We formalize PD and PFI as statistical estimators of ground truth estimands rooted in the data generating process. We show that PD and PFI estimates deviate from this ground truth due to statistical biases, model variance and Monte Carlo approximation errors. To account for model variance in PD and PFI estimation, we propose the learner-PD and the learner-PFI based on model refits, and propose corrected variance and confidence interval estimators.