# Joint Shapley values: a measure of joint feature importance

arXiv: https://arxiv.org/abs/2107.11357 · HTML: https://mlsanity.com/paper/2107.11357/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 24 references that are in the map.
- First posted on arXiv 23 Jul 2021.

## Builds on (most cited first, 7 of 7)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Explainable Machine Learning in Deployment](https://mlsanity.com/paper/1909.06342/) (2019)
- [The Shapley Taylor Interaction Index](https://mlsanity.com/paper/1902.05622/) (2019)
- [Asymmetric Shapley values: incorporating causal knowledge into model-agnostic explainability](https://mlsanity.com/paper/1910.06358/) (2019)
- [Sampling Permutations for Shapley Value Estimation](https://mlsanity.com/paper/2104.12199/) (2021)
- [Efficient nonparametric statistical inference on population feature importance using Shapley values](https://mlsanity.com/paper/2006.09481/) (2020)
- [High Dimensional Model Explanations: an Axiomatic Approach](https://mlsanity.com/paper/2006.08969/) (2020)

## Built on by (most cited first, 15 of 15)

- [Measuring the Effect of Training Data on Deep Learning Predictions via Randomized Experiments](https://mlsanity.com/paper/2206.10013/) (2022)
- [shapiq: Shapley Interactions for Machine Learning](https://mlsanity.com/paper/2410.01649/) (2024)
- [KernelSHAP-IQ: Weighted Least-Square Optimization for Shapley Interactions](https://mlsanity.com/paper/2405.10852/) (2024)
- [Using Shapley Values and Variational Autoencoders to Explain Predictive Models with Dependent Mixed Features](https://mlsanity.com/paper/2111.13507/) (2021)
- [Unifying local and global model explanations by functional decomposition of low dimensional structures](https://mlsanity.com/paper/2208.06151/) (2022)
- [Learning to Understand: Identifying Interactions via the M"obius Transform](https://mlsanity.com/paper/2402.02631/) (2024)
- [Unifying Feature-Based Explanations with Functional ANOVA and Cooperative Game Theory](https://mlsanity.com/paper/2412.17152/) (2024)
- [GeoShapley: A Game Theory Approach to Measuring Spatial Effects in Machine Learning Models](https://mlsanity.com/paper/2312.03675/) (2023)
- [Towards Attributions of Input Variables in a Coalition](https://mlsanity.com/paper/2309.13411/) (2023)
- [Can Moran Eigenvectors Improve Machine Learning of Spatial Data? Insights from Synthetic Data Validation](https://mlsanity.com/paper/2504.12450/) (2025)
- [Succinct Interaction-Aware Explanations](https://mlsanity.com/paper/2402.05566/) (2024)
- [Saliency Methods are Encoders: Analysing Logical Relations Towards Interpretation](https://mlsanity.com/paper/2412.16204/) (2024)
- [The Geography of Transportation Cybersecurity: Visitor Flows, Industry Clusters, and Spatial Dynamics](https://mlsanity.com/paper/2505.08822/) (2025)
- [SX-GeoTree: Self-eXplaining Geospatial Regression Tree Incorporating the Spatial Similarity of Feature Attributions](https://mlsanity.com/paper/2511.19845/) (2025)
- [Attribution Upsampling should Redistribute, Not Interpolate](https://mlsanity.com/paper/2603.16067/) (2026)

## Abstract (arXiv)

The Shapley value is one of the most widely used measures of feature importance partly as it measures a feature's average effect on a model's prediction. We introduce joint Shapley values, which directly extend Shapley's axioms and intuitions: joint Shapley values measure a set of features' average contribution to a model's prediction. We prove the uniqueness of joint Shapley values, for any order of explanation. Results for games show that joint Shapley values present different insights from existing interaction indices, which assess the effect of a feature within a set of features. The joint Shapley values provide intuitive results in ML attribution problems. With binary features, we present a presence-adjusted global value that is more consistent with local intuitions than the usual approach.