# Explaining Predictive Uncertainty with Information Theoretic Shapley Values

arXiv: https://arxiv.org/abs/2306.05724 · HTML: https://mlsanity.com/paper/2306.05724/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 58 of its 96 references that are in the map.
- First posted on arXiv 9 Jun 2023.

## Builds on (most cited first, 30 of 58)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Sanity Checks for Saliency Maps](https://mlsanity.com/paper/1810.03292/) (2018)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [The Mythos of Model Interpretability](https://mlsanity.com/paper/1606.03490/) (2016)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [Deep Exploration via Bootstrapped DQN](https://mlsanity.com/paper/1602.04621/) (2016)
- [BatchBALD: Efficient and Diverse Batch Acquisition for Deep Bayesian Active Learning](https://mlsanity.com/paper/1906.08158/) (2019)
- [The many Shapley values for model explanation](https://mlsanity.com/paper/1908.08474/) (2019)
- [Learning to Explain: An Information-Theoretic Perspective on Model Interpretation](https://mlsanity.com/paper/1802.07814/) (2018)
- [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)
- [Model-Agnostic Counterfactual Explanations for Consequential Decisions](https://mlsanity.com/paper/1905.11190/) (2019)
- [Understanding Measures of Uncertainty for Adversarial Example Detection](https://mlsanity.com/paper/1803.08533/) (2018)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Feature relevance quantification in explainable AI: A causal problem](https://mlsanity.com/paper/1910.13413/) (2019)
- [Improving KernelSHAP: Practical Shapley Value Estimation via Linear Regression](https://mlsanity.com/paper/2012.01536/) (2020)
- [Testing for Outliers with Conformal p-values](https://mlsanity.com/paper/2104.08279/) (2021)
- [True to the Model or True to the Data?](https://mlsanity.com/paper/2006.16234/) (2020)

## Built on by (most cited first, 30 of 36)

- [shapiq: Shapley Interactions for Machine Learning](https://mlsanity.com/paper/2410.01649/) (2024)
- [Uncertainty in Graph Neural Networks: A Survey](https://mlsanity.com/paper/2403.07185/) (2024)
- [Anthropomimetic Uncertainty: What Verbalized Uncertainty in Language Models is Missing](https://mlsanity.com/paper/2507.10587/) (2025)
- [Explaining Bayesian Optimization by Shapley Values Facilitates Human-AI Collaboration](https://mlsanity.com/paper/2403.04629/) (2024)
- [Explaining Predictive Uncertainty by Exposing Second-Order Effects](https://mlsanity.com/paper/2401.17441/) (2024)
- [Uncertainty quantification for probabilistic machine learning in earth observation using conformal prediction](https://mlsanity.com/paper/2401.06421/) (2024)
- [Model-agnostic variable importance for predictive uncertainty: an entropy-based approach](https://mlsanity.com/paper/2310.12842/) (2023)
- [Unveil Sources of Uncertainty: Feature Contribution to Conformal Prediction Intervals](https://mlsanity.com/paper/2505.13118/) (2025)
- [Identifying Drivers of Predictive Aleatoric Uncertainty](https://mlsanity.com/paper/2312.07252/) (2023)
- [X-Fake: Juggling Utility Evaluation and Explanation of Simulated SAR Images](https://mlsanity.com/paper/2407.19436/) (2024)
- [UbiQTree: Uncertainty Quantification in XAI with Tree Ensembles](https://mlsanity.com/paper/2508.09639/) (2025)
- [LLM Performance Predictors: Learning When to Escalate in Hybrid Human-AI Moderation Systems](https://mlsanity.com/paper/2601.07006/) (2026)
- [Investigating the Impact of Model Instability on Explanations and Uncertainty](https://mlsanity.com/paper/2402.13006/) (2024)
- [Sufficient and Necessary Explanations (and What Lies in Between)](https://mlsanity.com/paper/2409.20427/) (2024)
- [Beyond Shapley Values: Cooperative Games for the Interpretation of Machine Learning Models](https://mlsanity.com/paper/2506.13900/) (2025)
- [A Comprehensive Study of Shapley Value in Data Analytics](https://mlsanity.com/paper/2412.01460/) (2024)
- [Segmentation, Classification and Interpretation of Breast Cancer Medical Images using Human-in-the-Loop Machine Learning](https://mlsanity.com/paper/2403.20112/) (2024)
- [On Using the Shapley Value for Anomaly Localization: A Statistical Investigation](https://mlsanity.com/paper/2507.21023/) (2025)
- [A Systematic Review of Conformal Inference Procedures for Treatment Effect Estimation: Methods and Challenges](https://mlsanity.com/paper/2509.21660/) (2025)
- [Uncertainty-Aware Multimodal Learning via Conformal Shapley Intervals](https://mlsanity.com/paper/2602.00171/) (2026)
- [Conceptualizing Uncertainty: A Concept-based Approach to Explaining Uncertainty](https://mlsanity.com/paper/2503.03443/) (2025)
- [Understanding Action Effects through Instrumental Empowerment in Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2508.15652/) (2025)
- [φ-Table: A Statistical Explanation for Global SHAP](https://mlsanity.com/paper/2512.07578/) (2025)
- [UbiQVision: Quantifying Uncertainty in XAI for Image Recognition](https://mlsanity.com/paper/2512.20288/) (2025)
- [How important are the genes to explain the outcome - the asymmetric Shapley value as an honest importance metric for high-dimensional features](https://mlsanity.com/paper/2603.05317/) (2026)
- [EviATTA: Evidential Active Test-Time Adaptation for Medical Segment Anything Models](https://mlsanity.com/paper/2603.14666/) (2026)
- [No Single Metric Tells the Whole Story: A Multi-Dimensional Evaluation Framework for Uncertainty Attributions](https://mlsanity.com/paper/2603.24524/) (2026)
- [Quantifying and Understanding Uncertainty in Large Reasoning Models](https://mlsanity.com/paper/2604.13395/) (2026)
- [RoSHAP: A Distributional Framework and Robust Metric for Stable Feature Attribution](https://mlsanity.com/paper/2605.15154/) (2026)
- [Verified SHAP: Provable Bounds for Exact Shapley Values of Neural Networks](https://mlsanity.com/paper/2605.24084/) (2026)

## Abstract (arXiv)

Researchers in explainable artificial intelligence have developed numerous methods for helping users understand the predictions of complex supervised learning models. By contrast, explaining the textituncertainty of model outputs has received relatively little attention. We adapt the popular Shapley value framework to explain various types of predictive uncertainty, quantifying each feature's contribution to the conditional entropy of individual model outputs. We consider games with modified characteristic functions and find deep connections between the resulting Shapley values and fundamental quantities from information theory and conditional independence testing. We outline inference procedures for finite sample error rate control with provable guarantees, and implement efficient algorithms that perform well in a range of experiments on real and simulated data. Our method has applications to covariate shift detection, active learning, feature selection, and active feature-value acquisition.