# Faith-Shap: The Faithful Shapley Interaction Index

arXiv: https://arxiv.org/abs/2203.00870 · HTML: https://mlsanity.com/paper/2203.00870/

## Key facts

- Cited by 69 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 55 references that are in the map.
- First posted on arXiv 2 Mar 2022.

## Builds on (most cited first, 18 of 18)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Data Shapley: Equitable Valuation of Data for Machine Learning](https://mlsanity.com/paper/1904.02868/) (2019)
- [The many Shapley values for model explanation](https://mlsanity.com/paper/1908.08474/) (2019)
- [Towards Efficient Data Valuation Based on the Shapley Value](https://mlsanity.com/paper/1902.10275/) (2019)
- [On Completeness-aware Concept-Based Explanations in Deep Neural Networks](https://mlsanity.com/paper/1910.07969/) (2019)
- [Improving KernelSHAP: Practical Shapley Value Estimation via Linear Regression](https://mlsanity.com/paper/2012.01536/) (2020)
- [The Shapley Taylor Interaction Index](https://mlsanity.com/paper/1902.05622/) (2019)
- [True to the Model or True to the Data?](https://mlsanity.com/paper/2006.16234/) (2020)
- [Explaining Explanations: Axiomatic Feature Interactions for Deep Networks](https://mlsanity.com/paper/2002.04138/) (2020)
- [Sampling Permutations for Shapley Value Estimation](https://mlsanity.com/paper/2104.12199/) (2021)
- [Model Agnostic Supervised Local Explanations](https://mlsanity.com/paper/1807.02910/) (2018)
- [Detecting Statistical Interactions from Neural Network Weights](https://mlsanity.com/paper/1705.04977/) (2017)
- [How does this interaction affect me? Interpretable attribution for feature interactions](https://mlsanity.com/paper/2006.10965/) (2020)
- [Hierarchical interpretations for neural network predictions](https://mlsanity.com/paper/1806.05337/) (2018)
- [Connecting Attributions and QA Model Behavior on Realistic Counterfactuals](https://mlsanity.com/paper/2104.04515/) (2021)

## Built on by (most cited first, 30 of 69)

- [Training Data Influence Analysis and Estimation: A Survey](https://mlsanity.com/paper/2212.04612/) (2022)
- [From Shapley Values to Generalized Additive Models and back](https://mlsanity.com/paper/2209.04012/) (2022)
- [Explainable and Interpretable Multimodal Large Language Models: A Comprehensive Survey](https://mlsanity.com/paper/2412.02104/) (2024)
- [SHAP-IQ: Unified Approximation of any-order Shapley Interactions](https://mlsanity.com/paper/2303.01179/) (2023)
- [shapiq: Shapley Interactions for Machine Learning](https://mlsanity.com/paper/2410.01649/) (2024)
- [Approximating the Shapley Value without Marginal Contributions](https://mlsanity.com/paper/2302.00736/) (2023)
- [Beyond TreeSHAP: Efficient Computation of Any-Order Shapley Interactions for Tree Ensembles](https://mlsanity.com/paper/2401.12069/) (2024)
- [Does a Neural Network Really Encode Symbolic Concepts?](https://mlsanity.com/paper/2302.13080/) (2023)
- [KernelSHAP-IQ: Weighted Least-Square Optimization for Shapley Interactions](https://mlsanity.com/paper/2405.10852/) (2024)
- [Unifying local and global model explanations by functional decomposition of low dimensional structures](https://mlsanity.com/paper/2208.06151/) (2022)
- [Measuring Cross-Modal Interactions in Multimodal Models](https://mlsanity.com/paper/2412.15828/) (2024)
- [ProxySPEX: Inference-Efficient Interpretability via Sparse Feature Interactions in LLMs](https://mlsanity.com/paper/2505.17495/) (2025)
- [CausalArmor: Efficient Indirect Prompt Injection Guardrails via Causal Attribution](https://mlsanity.com/paper/2602.07918/) (2026)
- [Learning to Understand: Identifying Interactions via the M"obius Transform](https://mlsanity.com/paper/2402.02631/) (2024)
- [Exact Computation of Any-Order Shapley Interactions for Graph Neural Networks](https://mlsanity.com/paper/2501.16944/) (2025)
- [Technical Note: Defining and Quantifying AND-OR Interactions for Faithful and Concise Explanation of DNNs](https://mlsanity.com/paper/2304.13312/) (2023)
- [SPEX: Scaling Feature Interaction Explanations for LLMs](https://mlsanity.com/paper/2502.13870/) (2025)
- [Explaining Graph Neural Networks via Structure-aware Interaction Index](https://mlsanity.com/paper/2405.14352/) (2024)
- [Explainability in Practice: A Survey of Explainable NLP Across Various Domains](https://mlsanity.com/paper/2502.00837/) (2025)
- [Sample based Explanations via Generalized Representers](https://mlsanity.com/paper/2310.18526/) (2023)
- [Unifying Feature-Based Explanations with Functional ANOVA and Cooperative Game Theory](https://mlsanity.com/paper/2412.17152/) (2024)
- [Where We Have Arrived in Proving the Emergence of Sparse Symbolic Concepts in AI Models](https://mlsanity.com/paper/2305.01939/) (2023)
- [MultiSHAP: A Shapley-Based Framework for Explaining Cross-Modal Interactions in Multimodal AI Models](https://mlsanity.com/paper/2508.00576/) (2025)
- [PolySHAP: Extending KernelSHAP with Interaction-Informed Polynomial Regression](https://mlsanity.com/paper/2601.18608/) (2026)
- [Explaining Similarity in Vision-Language Encoders with Weighted Banzhaf Interactions](https://mlsanity.com/paper/2508.05430/) (2025)
- [Four Axiomatic Characterizations of the Integrated Gradients Attribution Method](https://mlsanity.com/paper/2306.13753/) (2023)
- [Interpretable Machine Learning for Survival Analysis](https://mlsanity.com/paper/2403.10250/) (2024)
- [On the Inflation of KNN-Shapley Value](https://mlsanity.com/paper/2405.17489/) (2024)
- [SHAP zero Explains Biological Sequence Models with Near-zero Marginal Cost for Future Queries](https://mlsanity.com/paper/2410.19236/) (2024)
- [HyperSHAP: Shapley Values and Interactions for Explaining Hyperparameter Optimization](https://mlsanity.com/paper/2502.01276/) (2025)

## Abstract (arXiv)

Shapley values, which were originally designed to assign attributions to individual players in coalition games, have become a commonly used approach in explainable machine learning to provide attributions to input features for black-box machine learning models. A key attraction of Shapley values is that they uniquely satisfy a very natural set of axiomatic properties. However, extending the Shapley value to assigning attributions to interactions rather than individual players, an interaction index, is non-trivial: as the natural set of axioms for the original Shapley values, extended to the context of interactions, no longer specify a unique interaction index. Many proposals thus introduce additional less ''natural'' axioms, while sacrificing the key axiom of efficiency, in order to obtain unique interaction indices. In this work, rather than introduce additional conflicting axioms, we adopt the viewpoint of Shapley values as coefficients of the most faithful linear approximation to the pseudo-Boolean coalition game value function. By extending linear to ℓ-order polynomial approximations, we can then define the general family of faithful interaction indices. We show that by additionally requiring the faithful interaction indices to satisfy interaction-extensions of the standard individual Shapley axioms (dummy, symmetry, linearity, and efficiency), we obtain a unique Faithful Shapley Interaction index, which we denote Faith-Shap, as a natural generalization of the Shapley value to interactions. We then provide some illustrative contrasts of Faith-Shap with previously proposed interaction indices, and further investigate some of its interesting algebraic properties. We further show the computational efficiency of computing Faith-Shap, together with some additional qualitative insights, via some illustrative experiments.