# A Hierarchy of Entropy-Shapley Games for Multivariate Predictive Uncertainty

arXiv: https://arxiv.org/abs/2609.35217 · HTML: https://mlsanity.com/paper/2609.35217/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 36 of its 56 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 30 of 36)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Chronos: Learning the Language of Time Series](https://mlsanity.com/paper/2403.07815/) (2024)
- [A decoder-only foundation model for time-series forecasting](https://mlsanity.com/paper/2310.10688/) (2023)
- [Unified Training of Universal Time Series Forecasting Transformers](https://mlsanity.com/paper/2402.02592/) (2024)
- [All Models are Wrong, but Many are Useful: Learning a Variable's Importance by Studying an Entire Class of Prediction Models Simultaneously](https://mlsanity.com/paper/1801.01489/) (2018)
- [The many Shapley values for model explanation](https://mlsanity.com/paper/1908.08474/) (2019)
- [Consistent Individualized Feature Attribution for Tree Ensembles](https://mlsanity.com/paper/1802.03888/) (2018)
- [MultiPath++: Efficient Information Fusion and Trajectory Aggregation for Behavior Prediction](https://mlsanity.com/paper/2111.14973/) (2021)
- [Explaining individual predictions when features are dependent: More accurate approximations to Shapley values](https://mlsanity.com/paper/1903.10464/) (2019)
- [Algorithms to estimate Shapley value feature attributions](https://mlsanity.com/paper/2207.07605/) (2022)
- [Neural networks for post-processing ensemble weather forecasts](https://mlsanity.com/paper/1805.09091/) (2018)
- [NGBoost: Natural Gradient Boosting for Probabilistic Prediction](https://mlsanity.com/paper/1910.03225/) (2019)
- [Shapley explainability on the data manifold](https://mlsanity.com/paper/2006.01272/) (2020)
- [TimeSHAP: Explaining Recurrent Models through Sequence Perturbations](https://mlsanity.com/paper/2012.00073/) (2020)
- [Explaining Predictive Uncertainty with Information Theoretic Shapley Values](https://mlsanity.com/paper/2306.05724/) (2023)
- [Efficient multivariate entropy estimation via k-nearest neighbour distances](https://mlsanity.com/paper/1606.00304/) (2016)
- [SurvSHAP(t): Time-dependent explanations of machine learning survival models](https://mlsanity.com/paper/2208.11080/) (2022)
- [WindowSHAP: An Efficient Framework for Explaining Time-series Classifiers based on Shapley Values](https://mlsanity.com/paper/2211.06507/) (2022)
- [Don't be fooled: label leakage in explanation methods and the importance of their quantitative evaluation](https://mlsanity.com/paper/2302.12893/) (2023)
- [Unifying Feature-Based Explanations with Functional ANOVA and Cooperative Game Theory](https://mlsanity.com/paper/2412.17152/) (2024)
- [Sensitivity analysis for multidimensional and functional outputs](https://mlsanity.com/paper/1311.1797/) (2013)
- [Model-agnostic variable importance for predictive uncertainty: an entropy-based approach](https://mlsanity.com/paper/2310.12842/) (2023)
- [Uncertainty Propagation in XAI: A Comparison of Analytical and Empirical Estimators](https://mlsanity.com/paper/2504.03736/) (2025)

## Abstract (arXiv)

Modern probabilistic machine learning models increasingly produce multivariate outputs with complex dependence structure, from multi-step time-series forecasts to sample path predictions. Understanding which input features drive the predictive uncertainty is important for risk-aware decisions, model diagnostics, and deciding whether the uncertainty should be mitigated or hedged against. This attribution problem requires a choice of how dependencies between output components are treated. Existing approaches reduce the output to a scalar through aggregation or projection before attribution, thereby obscuring whether features affect marginal uncertainty, dependence structure, or both, while component-wise analyses can miss dependence effects entirely. We close this gap by introducing a hierarchy of three entropy-based Shapley games that make this output-side choice explicit for any ordered multivariate outcome, ranging from per-component marginal entropy to fully joint entropy. The hierarchy isolates a cross-component attribution term that captures how each feature shifts the dependence between output components, a quantity invisible to component-wise methods. We establish a chain-rule decomposition of the joint attribution and characterize the cross-component term through conditional total correlation, providing both closed-form and sample-based estimators. Finally, we demonstrate how the framework captures differences in learned joint structure across probabilistic models from distributional regression to a zero-shot time series foundation model.