# The Shapley Taylor Interaction Index

arXiv: https://arxiv.org/abs/1902.05622 · HTML: https://mlsanity.com/paper/1902.05622/

## Key facts

- Cited by 143 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2019.

## Built on by (most cited first, 30 of 143)

- [Training Data Influence Analysis and Estimation: A Survey](https://mlsanity.com/paper/2212.04612/) (2022)
- [The Shapley Value in Machine Learning](https://mlsanity.com/paper/2202.05594/) (2022)
- [Neuron-level Interpretation of Deep NLP Models: A Survey](https://mlsanity.com/paper/2108.13138/) (2021)
- [Faith-Shap: The Faithful Shapley Interaction Index](https://mlsanity.com/paper/2203.00870/) (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)
- [Discovering and Explaining the Representation Bottleneck of DNNs](https://mlsanity.com/paper/2111.06236/) (2021)
- [Defining and Quantifying the Emergence of Sparse Concepts in DNNs](https://mlsanity.com/paper/2111.06206/) (2021)
- [Beyond TreeSHAP: Efficient Computation of Any-Order Shapley Interactions for Tree Ensembles](https://mlsanity.com/paper/2401.12069/) (2024)
- [Sparse Interaction Additive Networks via Feature Interaction Detection and Sparse Selection](https://mlsanity.com/paper/2209.09326/) (2022)
- [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)
- [Discovering Salient Neurons in Deep NLP Models](https://mlsanity.com/paper/2206.13288/) (2022)
- [Stochastic Amortization: A Unified Approach to Accelerate Feature and Data Attribution](https://mlsanity.com/paper/2401.15866/) (2024)
- [Unifying local and global model explanations by functional decomposition of low dimensional structures](https://mlsanity.com/paper/2208.06151/) (2022)
- [PredDiff: Explanations and Interactions from Conditional Expectations](https://mlsanity.com/paper/2102.13519/) (2021)
- [A Unified Game-Theoretic Interpretation of Adversarial Robustness](https://mlsanity.com/paper/2111.03536/) (2021)
- [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)
- [MIMIC-IF: Interpretability and Fairness Evaluation of Deep Learning Models on MIMIC-IV Dataset](https://mlsanity.com/paper/2102.06761/) (2021)
- [Joint Shapley values: a measure of joint feature importance](https://mlsanity.com/paper/2107.11357/) (2021)
- [Explainable Deep Learning in Healthcare: A Methodological Survey from an Attribution View](https://mlsanity.com/paper/2112.02625/) (2021)
- [Explanations of Black-Box Models based on Directional Feature Interactions](https://mlsanity.com/paper/2304.07670/) (2023)
- [Efficient XAI Techniques: A Taxonomic Survey](https://mlsanity.com/paper/2302.03225/) (2023)
- [Learning to Understand: Identifying Interactions via the M"obius Transform](https://mlsanity.com/paper/2402.02631/) (2024)
- [Towards the Dynamics of a DNN Learning Symbolic Interactions](https://mlsanity.com/paper/2407.19198/) (2024)
- [Exact Computation of Any-Order Shapley Interactions for Graph Neural Networks](https://mlsanity.com/paper/2501.16944/) (2025)
- [Interpreting Representation Quality of DNNs for 3D Point Cloud Processing](https://mlsanity.com/paper/2111.03549/) (2021)

## Abstract (arXiv)

The attribution problem, that is the problem of attributing a model's prediction to its base features, is well-studied. We extend the notion of attribution to also apply to feature interactions. The Shapley value is a commonly used method to attribute a model's prediction to its base features. We propose a generalization of the Shapley value called Shapley-Taylor index that attributes the model's prediction to interactions of subsets of features up to some size k. The method is analogous to how the truncated Taylor Series decomposes the function value at a certain point using its derivatives at a different point. In fact, we show that the Shapley Taylor index is equal to the Taylor Series of the multilinear extension of the set-theoretic behavior of the model. We axiomatize this method using the standard Shapley axioms -- linearity, dummy, symmetry and efficiency -- and an additional axiom that we call the interaction distribution axiom. This new axiom explicitly characterizes how interactions are distributed for a class of functions that model pure interaction. We contrast the Shapley-Taylor index against the previously proposed Shapley Interaction index (cf. [9]) from the cooperative game theory literature. We also apply the Shapley Taylor index to three models and identify interesting qualitative insights.