# Efficient computation and analysis of distributional Shapley values

arXiv: https://arxiv.org/abs/2007.01357 · HTML: https://mlsanity.com/paper/2007.01357/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2020.

## Built on by (most cited first, 30 of 38)

- [Beta Shapley: a Unified and Noise-reduced Data Valuation Framework for Machine Learning](https://mlsanity.com/paper/2110.14049/) (2021)
- [Data Banzhaf: A Robust Data Valuation Framework for Machine Learning](https://mlsanity.com/paper/2205.15466/) (2022)
- [The Shapley Value in Machine Learning](https://mlsanity.com/paper/2202.05594/) (2022)
- [Data Shapley in One Training Run](https://mlsanity.com/paper/2406.11011/) (2024)
- [Shapley values for feature selection: The good, the bad, and the axioms](https://mlsanity.com/paper/2102.10936/) (2021)
- [Synthetic Datasets for Autonomous Driving: A Survey](https://mlsanity.com/paper/2304.12205/) (2023)
- [Data-OOB: Out-of-bag Estimate as a Simple and Efficient Data Value](https://mlsanity.com/paper/2304.07718/) (2023)
- [WeightedSHAP: analyzing and improving Shapley based feature attributions](https://mlsanity.com/paper/2209.13429/) (2022)
- [Data Valuation Without Training of a Model](https://mlsanity.com/paper/2301.00930/) (2023)
- [CS-Shapley: Class-wise Shapley Values for Data Valuation in Classification](https://mlsanity.com/paper/2211.06800/) (2022)
- [Interpretable Data-Based Explanations for Fairness Debugging](https://mlsanity.com/paper/2112.09745/) (2021)
- [Rethinking Data Shapley for Data Selection Tasks: Misleads and Merits](https://mlsanity.com/paper/2405.03875/) (2024)
- [Improving Fairness for Data Valuation in Horizontal Federated Learning](https://mlsanity.com/paper/2109.09046/) (2021)
- [Fair and efficient contribution valuation for vertical federated learning](https://mlsanity.com/paper/2201.02658/) (2022)
- [Data Valuation and Detections in Federated Learning](https://mlsanity.com/paper/2311.05304/) (2023)
- [DU-Shapley: A Shapley Value Proxy for Efficient Dataset Valuation](https://mlsanity.com/paper/2306.02071/) (2023)
- [Data Acquisition: A New Frontier in Data-centric AI](https://mlsanity.com/paper/2311.13712/) (2023)
- [Differentially Private Shapley Values for Data Evaluation](https://mlsanity.com/paper/2206.00511/) (2022)
- [BWS: Best Window Selection Based on Sample Scores for Data Pruning across Broad Ranges](https://mlsanity.com/paper/2406.03057/) (2024)
- [SHAP values via sparse Fourier representation](https://mlsanity.com/paper/2410.06300/) (2024)
- [Addressing Budget Allocation and Revenue Allocation in Data Market Environments Using an Adaptive Sampling Algorithm](https://mlsanity.com/paper/2306.02543/) (2023)
- [DIVINE: Diverse Influential Training Points for Data Visualization and Model Refinement](https://mlsanity.com/paper/2107.05978/) (2021)
- [The Thousand Faces of Explainable AI Along the Machine Learning Life Cycle: Industrial Reality and Current State of Research](https://mlsanity.com/paper/2310.07882/) (2023)
- [2D-OOB: Attributing Data Contribution Through Joint Valuation Framework](https://mlsanity.com/paper/2408.03572/) (2024)
- [Beyond Demographic Parity: Redefining Equal Treatment](https://mlsanity.com/paper/2303.08040/) (2023)
- [Farzi Data: Autoregressive Data Distillation](https://mlsanity.com/paper/2310.09983/) (2023)
- [Inferring feature importance with uncertainties in high-dimensional data](https://mlsanity.com/paper/2109.00855/) (2021)
- [EcoVal: An Efficient Data Valuation Framework for Machine Learning](https://mlsanity.com/paper/2402.09288/) (2024)
- [A Comprehensive Study of Shapley Value in Data Analytics](https://mlsanity.com/paper/2412.01460/) (2024)
- [FW-Shapley: Real-time Estimation of Weighted Shapley Values](https://mlsanity.com/paper/2503.06602/) (2025)

## Abstract (arXiv)

Distributional data Shapley value (DShapley) has recently been proposed as a principled framework to quantify the contribution of individual datum in machine learning. DShapley develops the foundational game theory concept of Shapley values into a statistical framework and can be applied to identify data points that are useful (or harmful) to a learning algorithm. Estimating DShapley is computationally expensive, however, and this can be a major challenge to using it in practice. Moreover, there has been little mathematical analyses of how this value depends on data characteristics. In this paper, we derive the first analytic expressions for DShapley for the canonical problems of linear regression, binary classification, and non-parametric density estimation. These analytic forms provide new algorithms to estimate DShapley that are several orders of magnitude faster than previous state-of-the-art methods. Furthermore, our formulas are directly interpretable and provide quantitative insights into how the value varies for different types of data. We demonstrate the practical efficacy of our approach on multiple real and synthetic datasets.