# Shapley Curves: A Smoothing Perspective

arXiv: https://arxiv.org/abs/2211.13289 · HTML: https://mlsanity.com/paper/2211.13289/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 70 references that are in the map.
- First posted on arXiv 23 Nov 2022.

## Builds on (most cited first, 24 of 24)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [The many Shapley values for model explanation](https://mlsanity.com/paper/1908.08474/) (2019)
- [Explaining by Removing: A Unified Framework for Model Explanation](https://mlsanity.com/paper/2011.14878/) (2020)
- [Explaining individual predictions when features are dependent: More accurate approximations to Shapley values](https://mlsanity.com/paper/1903.10464/) (2019)
- [Improving KernelSHAP: Practical Shapley Value Estimation via Linear Regression](https://mlsanity.com/paper/2012.01536/) (2020)
- [FastSHAP: Real-Time Shapley Value Estimation](https://mlsanity.com/paper/2107.07436/) (2021)
- [Interpretable Machine Learning -- A Brief History, State-of-the-Art and Challenges](https://mlsanity.com/paper/2010.09337/) (2020)
- [Algorithms to estimate Shapley value feature attributions](https://mlsanity.com/paper/2207.07605/) (2022)
- [Shapley explainability on the data manifold](https://mlsanity.com/paper/2006.01272/) (2020)
- [Consistency of random forests](https://mlsanity.com/paper/1405.2881/) (2014)
- [Efficient nonparametric statistical inference on population feature importance using Shapley values](https://mlsanity.com/paper/2006.09481/) (2020)
- [A Multilinear Sampling Algorithm to Estimate Shapley Values](https://mlsanity.com/paper/2010.12082/) (2020)
- [SHAFF: Fast and consistent SHApley eFfect estimates via random Forests](https://mlsanity.com/paper/2105.11724/) (2021)
- [Unifying local and global model explanations by functional decomposition of low dimensional structures](https://mlsanity.com/paper/2208.06151/) (2022)
- [Trees, forests, and impurity-based variable importance](https://mlsanity.com/paper/2001.04295/) (2020)
- [Decorrelated Variable Importance](https://mlsanity.com/paper/2111.10853/) (2021)
- [MDA for random forests: inconsistency, and a practical solution via the Sobol-MDA](https://mlsanity.com/paper/2102.13347/) (2021)
- [Explaining predictive models using Shapley values and non-parametric vine copulas](https://mlsanity.com/paper/2102.06416/) (2021)
- [Random Planted Forest: a directly interpretable tree ensemble](https://mlsanity.com/paper/2012.14563/) (2020)
- [Explainable AI for a No-Teardown Vehicle Component Cost Estimation: A Top-Down Approach](https://mlsanity.com/paper/2006.08828/) (2020)
- [Marginal integration for nonparametric causal inference](https://mlsanity.com/paper/1405.1868/) (2014)
- [Inferring feature importance with uncertainties in high-dimensional data](https://mlsanity.com/paper/2109.00855/) (2021)
- [A Data-driven Case-based Reasoning in Bankruptcy Prediction](https://mlsanity.com/paper/2211.00921/) (2022)

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

- [Interpretable Tabular Foundation Models via In-Context Kernel Regression](https://mlsanity.com/paper/2602.02162/) (2026)
- [Statistical Inference and Learning for Shapley Additive Explanations (SHAP)](https://mlsanity.com/paper/2602.10532/) (2026)

## Abstract (arXiv)

This paper fills the limited statistical understanding of Shapley values as a variable importance measure from a nonparametric (or smoothing) perspective. We introduce population-level textitShapley curves to measure the true variable importance, determined by the conditional expectation function and the distribution of covariates. Having defined the estimand, we derive minimax convergence rates and asymptotic normality under general conditions for the two leading estimation strategies. For finite sample inference, we propose a novel version of the wild bootstrap procedure tailored for capturing lower-order terms in the estimation of Shapley curves. Numerical studies confirm our theoretical findings, and an empirical application analyzes the determining factors of vehicle prices.