# Explaining the Uncertain: Stochastic Shapley Values for Gaussian Process Models

arXiv: https://arxiv.org/abs/2305.15167 · HTML: https://mlsanity.com/paper/2305.15167/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 52 references that are in the map.
- First posted on arXiv 24 May 2023.

## Builds on (most cited first, 23 of 23)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Consistent Individualized Feature Attribution for Tree Ensembles](https://mlsanity.com/paper/1802.03888/) (2018)
- [Explaining by Removing: A Unified Framework for Model Explanation](https://mlsanity.com/paper/2011.14878/) (2020)
- [Problems with Shapley-value-based explanations as feature importance measures](https://mlsanity.com/paper/2002.11097/) (2020)
- [Improving KernelSHAP: Practical Shapley Value Estimation via Linear Regression](https://mlsanity.com/paper/2012.01536/) (2020)
- [Uncertainty as a Form of Transparency: Measuring, Communicating, and Using Uncertainty](https://mlsanity.com/paper/2011.07586/) (2020)
- [True to the Model or True to the Data?](https://mlsanity.com/paper/2006.16234/) (2020)
- [Reliable Post hoc Explanations: Modeling Uncertainty in Explainability](https://mlsanity.com/paper/2008.05030/) (2020)
- [FastSHAP: Real-Time Shapley Value Estimation](https://mlsanity.com/paper/2107.07436/) (2021)
- [Asymmetric Shapley values: incorporating causal knowledge into model-agnostic explainability](https://mlsanity.com/paper/1910.06358/) (2019)
- [Causal Shapley Values: Exploiting Causal Knowledge to Explain Individual Predictions of Complex Models](https://mlsanity.com/paper/2011.01625/) (2020)
- [RKHS-SHAP: Shapley Values for Kernel Methods](https://mlsanity.com/paper/2110.09167/) (2021)
- [On Locality of Local Explanation Models](https://mlsanity.com/paper/2106.14648/) (2021)
- [Deconditional Downscaling with Gaussian Processes](https://mlsanity.com/paper/2105.12909/) (2021)
- [BayesIMP: Uncertainty Quantification for Causal Data Fusion](https://mlsanity.com/paper/2106.03477/) (2021)
- [Cohort Shapley value for algorithmic fairness](https://mlsanity.com/paper/2105.07168/) (2021)
- [Graphical Lasso and Thresholding: Equivalence and Closed-form Solutions](https://mlsanity.com/paper/1708.09479/) (2017)
- [Explaining Preferences with Shapley Values](https://mlsanity.com/paper/2205.13662/) (2022)
- [Boundary-Aware Uncertainty for Feature Attribution Explainers](https://mlsanity.com/paper/2210.02419/) (2022)
- [Variance Allocation and Shapley Value](https://mlsanity.com/paper/1606.09424/) (2016)
- [Causal versus Marginal Shapley Values for Robotic Lever Manipulation Controlled using Deep Reinforcement Learning](https://mlsanity.com/paper/2111.02936/) (2021)

## Built on by (most cited first, 19 of 19)

- [Looping in the Human Collaborative and Explainable Bayesian Optimization](https://mlsanity.com/paper/2310.17273/) (2023)
- [Integral Imprecise Probability Metrics](https://mlsanity.com/paper/2505.16156/) (2025)
- [SHAP Meets Tensor Networks: Provably Tractable Explanations with Parallelism](https://mlsanity.com/paper/2510.21599/) (2025)
- [Model-agnostic variable importance for predictive uncertainty: an entropy-based approach](https://mlsanity.com/paper/2310.12842/) (2023)
- [Exact Shapley Attributions in Quadratic-time for FANOVA Gaussian Processes](https://mlsanity.com/paper/2508.14499/) (2025)
- [Amortized Linear-time Exact Shapley Value for Product-Kernel Methods](https://mlsanity.com/paper/2505.16516/) (2025)
- [Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](https://mlsanity.com/paper/2602.01667/) (2026)
- [Explaining Probabilistic Models with Distributional Values](https://mlsanity.com/paper/2402.09947/) (2024)
- [Uncertainty Quantification as a Principled Foundation for Explainable Artificial Intelligence: A Case Study of Counterfactual Explanations](https://mlsanity.com/paper/2502.17007/) (2025)
- [Comparative Explanations: Explanation Guided Decision Making for Human-in-the-Loop Preference Selection](https://mlsanity.com/paper/2504.03744/) (2025)
- [Explainable Learning with Gaussian Processes](https://mlsanity.com/paper/2403.07072/) (2024)
- [Explanation Design in Strategic Learning: Sufficient Explanations that Induce Non-harmful Responses](https://mlsanity.com/paper/2502.04058/) (2025)
- [When Do Credal Sets Stabilize? Fixed-Point Theorems for Credal Set Updates](https://mlsanity.com/paper/2510.04769/) (2025)
- [Probabilistic Shapley Value Modeling and Inference](https://mlsanity.com/paper/2402.04211/) (2024)
- [Kernel Quantile Embeddings and Associated Probability Metrics](https://mlsanity.com/paper/2505.20433/) (2025)
- [Dynamic Hyperparameter Importance for Efficient Multi-Objective Optimization](https://mlsanity.com/paper/2601.03166/) (2026)
- [Verified SHAP: Provable Bounds for Exact Shapley Values of Neural Networks](https://mlsanity.com/paper/2605.24084/) (2026)
- [ShaplEIG: Bayesian Experimental Design for Shapley Value Estimation](https://mlsanity.com/paper/2606.02247/) (2026)
- [Explainable Uncertainty Estimation for Reliable Medical AI](https://mlsanity.com/paper/2608.28052/) (2026)

## Abstract (arXiv)

We present a novel approach for explaining Gaussian processes (GPs) that can utilize the full analytical covariance structure present in GPs. Our method is based on the popular solution concept of Shapley values extended to stochastic cooperative games, resulting in explanations that are random variables. The GP explanations generated using our approach satisfy similar favorable axioms to standard Shapley values and possess a tractable covariance function across features and data observations. This covariance allows for quantifying explanation uncertainties and studying the statistical dependencies between explanations. We further extend our framework to the problem of predictive explanation, and propose a Shapley prior over the explanation function to predict Shapley values for new data based on previously computed ones. Our extensive illustrations demonstrate the effectiveness of the proposed approach.