Shapley Uncertainty in ML Attributions

• Shapley Uncertainty is a measure that quantifies the variability in model attribution by evaluating variance, confidence intervals, and distributional ranges of Shapley values.
• It extends cooperative game theoretic frameworks to cover model evaluation, data valuation, and feature importance through methods like Monte Carlo sampling and regression-based approximations.
• Empirical studies in deep learning and clinical applications demonstrate its effectiveness in guiding feature selection and ensuring robust, trustworthy model interpretations.

Shapley Uncertainty refers to the quantification of variability or confidence associated with Shapley value attributions in cooperative game-theoretic frameworks applied to explainability, model evaluation, and data valuation. Shapley values uniquely decompose the effect (e.g., model prediction, loss function, utility, or performance metric) across features, data contributors, or generated outputs so as to satisfy combinatorial fairness axioms. Shapley Uncertainty extends this by providing rigorous measures—such as variance, confidence intervals, or distributional ranges—for these attributions in the presence of stochasticity, estimation limitations, or model/data variability. Recent research has formalized and studied Shapley uncertainty in diverse settings: deep neural networks, probabilistic data, ensembles of nearly-optimal models, information-theoretic decompositions, stochastic games, and multimodal learning. These developments enable trustworthy, robust, and meaningful interpretation of feature importance, data value, or generated output uncertainty in complex machine learning systems.

1. Formal Definitions and Theoretical Foundations

Shapley uncertainty can be cast in several precise forms, depending on the source of uncertainty:

• Stochastic Data or Models: If the underlying data or model is itself random, the canonical Shapley value becomes a random variable. For instance, given players (features, data sources) each contributing samples from a distribution, the random coalition utilities induce a Shapley value whose mean and variance quantify both expected contribution and uncertainty (Jia et al., 20 Jan 2026). The variance can be expressed in closed form:

$$\mathrm{Var}(\phi_i) = \sum_{S}c_S^2\,\mathrm{Var}(\Delta_{i,S}) + \sum_{S \neq T} c_S c_T\,\mathrm{Cov}(\Delta_{i,S},\Delta_{i,T}),$$

where $c_S$ is the standard Shapley weighting and $\Delta_{i,S}$ is the random marginal gain for coalition $S$.

• Distributional Uncertainty: When the probability law over the entity population is unknown or within a set $U$ (e.g., hyper-rectangles of possible product distributions), the SHAP score of a feature $\phi_i(P)$ varies with $P\in U$. The interval $[\phi_i^-, \phi_i^+]$ brackets the possible range of feature attributions under all plausible distributions, with sharp bounds achievable via enumeration or mathematical programming (Cifuentes et al., 2024).
• Sampling Approximation: In most practical scenarios, Shapley value estimation is performed via Monte Carlo or regression-based approximations (e.g., KernelSHAP). Here, the estimation error's standard error and confidence intervals provide a measure of uncertainty due to finite function evaluations. For instance, under unbiased KernelSHAP (Covert et al., 2020), confidence intervals can be derived using the asymptotic normality of the estimator:

$$\hat{\beta}_{n} \sim \mathcal{N}\left(\beta^{*}, \frac{1}{n}\Sigma_\beta\right)$$

with $\Sigma_\beta$ derived analytically.

• Model and Rashomon-set Variability: The Shapley Variable Importance Cloud (ShapleyVIC) aggregates Shapley attributions over a Rashomon set of nearly-optimal models. The between-model variance, pooled in a random-effects meta-analysis, yields a 95% prediction interval capturing both within- and across-model importance uncertainty (Ning et al., 2022).
• Information-Theoretic Decompositions: When the value function is defined as predictive uncertainty, e.g., local conditional entropy $c_S$0, the Shapley values themselves explain uncertainty attribution, and conformal prediction yields finite-sample confidence intervals over their variation on unseen data (Watson et al., 2023).
• Latent-variable Shapley Models: Variational Shapley Networks treat each feature's Shapley component as a latent Gaussian random variable, yielding closed-form credible intervals for feature importance directly in a single forward pass (Ketenci et al., 2024).

2. Algorithmic Estimation and Variance-Reduction Methods

Shapley uncertainty estimation employs a variety of algorithmic strategies, each carefully designed to preserve the fairness and interpretability guarantees of the Shapley framework:

• Monte Carlo Sampling: Classic Shapley estimation draws random permutations or coalitions, computes marginal contributions, and uses sample mean and variance to estimate the expectation and uncertainty. This paradigm extends to stochastic games, probabilistic entities, and data-Shapley problems, with further refinements such as pick-and-freeze and Double Monte Carlo variance estimators (Wu et al., 2024, Jia et al., 20 Jan 2026).
• Regression-based Approximations: KernelSHAP models the characteristic game as a weighted least-squares problem, allowing analytical computations of estimation variance and convergence-based stopping rules. Paired-sampling and control variates further reduce Monte Carlo variance by exploiting correlation with closed-form surrogate games (Covert et al., 2020, Goldwasser et al., 2023).
• Uncertainty Propagation and Moment Matching: In deep neural networks, the DASP estimator (Deep Approximate Shapley Propagation) propagates Gaussian input uncertainty (arising from masking random feature coalitions) through all layers using moment-matching, efficiently approximating both the mean and variance of feature relevance in polynomial time (Ancona et al., 2019).
• Meta-analytic Aggregation: When aggregating attributions across a model cloud, ShapleyVIC performs a random-effects meta-analysis to combine within-model standard errors and between-model variance into a single uncertainty interval (Ning et al., 2022).
• Quadratic-time Recursions: For special model architectures such as FANOVA Gaussian processes, closed-form Möbius decompositions and Newton’s identities yield O($c_S$1) algorithms to compute both the exact mean and variance of each local or global Shapley attribution (Mohammadi et al., 20 Aug 2025).
• Conformal Intervals: Conditional conformal predictive approaches produce finite-sample, distribution-free uncertainty intervals for instance-wise or modality-level Shapley values, ensuring guaranteed marginal or conditional coverage (Chandy et al., 30 Jan 2026, Watson et al., 2023).

3. Empirical Evaluation and Practical Impact

Numerous experimental studies demonstrate that reporting Shapley uncertainty—via variance, confidence intervals, or robust intervals—enables more reliable interpretation and use of feature attributions:

• In deep networks, DASP reaches sub-0.01 RMSE in $c_S$2 forward passes, whereas sampling-based methods require $c_S$3 passes for comparable accuracy (Ancona et al., 2019).
• The variability of feature importance under sampling, model, or population uncertainty can materially affect ranking and feature selection decisions. ShapleyVIC intervals often expose overlapping importance for middle-ranked features, which would be indistinguishable under point estimates, thus supporting more robust and trustworthy variable selection (Ning et al., 2022).
• In clinical segmentation, Shapley rank variance across cross-validation folds serves as an actionable proxy for model uncertainty, correlating strongly and negatively with segmentation Dice score. Cases with high rank-variance reliably flag unreliable segmentations, providing clinicians with interpretable cues for triage and quality control (Ren et al., 8 Dec 2025).
• In data valuation markets, Wald-style and U-statistic derived confidence intervals on Data Shapley enable buyers to audit seller price quotes for honesty and spot mislabeled data via out-of-CI deviations (Wu et al., 2024).
• In natural language generation, Shapley-based semantic entropy decomposition (using pairwise entailment and PSD kernels) quantifies output uncertainty in a way that more tightly predicts LLM answer correctness than thresholded clustering-based semantic entropy or predictive entropy (Zhu et al., 29 Jul 2025).

4. Connections to Information Theory, Fairness, and Explanation Reliability

Information-theoretic generalizations of Shapley uncertainty decompose the total predictive entropy or variance into fair, context-dependent feature attributions:

• For entropy-based value functions, the sum of feature entropy-Shapley contributions equals the total information gain or decrease in uncertainty from the features revealed, which directly relates to local mutual information (Watson et al., 2023).
• These frameworks tightly connect Shapley uncertainty with conditional independence: nonzero local conditional mutual information terms imply dependence, and context-specific (i.e., instance-dependent) attributions witness unique feature interactions not captured by global metrics (Watson et al., 2023).
• Shapley uncertainty is uniquely axiomatically justified as an uncertainty metric for correlated outputs or semantic spaces, satisfying minimality, maximality, and consistency properties not always preserved by naive entropy or clustering-based measures (Zhu et al., 29 Jul 2025).
• For variance-based (global) attributions, FANOVA kernels provide fairness and efficiency, guaranteeing that attributions sum to total explained variance and fully capture all interaction effects (unlike marginal-only indices) (Mohammadi et al., 20 Aug 2025).

5. Domain-Specific Extensions and Contemporary Research Directions

Contemporary research extends and specializes Shapley uncertainty across a variety of applications and machine learning paradigms:

• Multimodal learning: Conformal Shapley intervals yield feature-wise, input-conditioned uncertainty bars for each modality, facilitating confident and near-optimal modality selection with finite-sample coverage guarantees (Chandy et al., 30 Jan 2026).
• Self-explaining probabilistic Shapley models: By parameterizing the distribution of per-feature Shapley contributions, e.g., as conditional Gaussians in Variational Shapley Networks, both mean and credible intervals of importance are reported directly in one forward pass, providing reliability for clinical and time-critical applications (Ketenci et al., 2024).
• Stochastic data sharing and federated valuation: Probabilistic Shapley formulations and stratified pooled estimators address the scenario where each contributor owns a data distribution, not fixed entities, capturing uncertainty arising from sampling, player heterogeneity, and federated dynamics (Jia et al., 20 Jan 2026).
• Contrast-level and medical imaging explainability: Shapley ranking variance as a global, model-agnostic uncertainty metric closely analogizes radiologist inter-rater variability, bridging the gap between algorithmic reliability and clinical reasoning (Ren et al., 8 Dec 2025).
• Fairness and robustness analysis: The length of Shapley score intervals or their overlap across features directly indicates the instability or fragility of importance rankings, compelling practitioners to interpret attribution intervals rather than point values alone in risk-sensitive applications (Cifuentes et al., 2024).

6. Limitations, Open Problems, and Future Directions

Notwithstanding broad progress, crucial issues remain:

• Scalability: Exact computation of Shapley intervals or variances is exponential in the number of features for general models; while quadratic-time solutions exist for certain structures (e.g., FANOVA GPs), most domains require variance-reduced sampling or moment-matching approximations (Mohammadi et al., 20 Aug 2025, Ancona et al., 2019).
• High-dimensional coalitions: Most methods become computationally intractable for $c_S$4, necessitating approximate Shapley solvers, linear regression surrogates, or combinatorial sampling (Covert et al., 2020).
• Off-manifold bias: In counterfactual or conditional expectation estimation, off-support sampling may introduce poorly calibrated uncertainty due to model misfit (Watson et al., 2023).
• Distributional vs. sampling uncertainty: Most uncertainty metrics focus on sampling or model variability; integrating epistemic, distributional, and group fairness notions remains an active area (Ning et al., 2022).
• Conditional coverage and calibration: Conformal intervals guarantee finite-sample marginal and, under regularity, conditional coverage, but further accuracy relies on calibration set representativity, choice of kernel or quantile regressor, and model complexity (Chandy et al., 30 Jan 2026).
• Axiomatic uniqueness: Shapley-based uncertainty metrics have been shown to be uniquely characterized by coalition-based entropy decompositions satisfying a minimal set of desirable axioms; alternative approaches may fail under certain semantic correlations (Zhu et al., 29 Jul 2025).

These unresolved challenges motivate development of scalable, model-structured, and theory-grounded approximations, integration with causal and interventional methods, and robust reporting standards for attribution uncertainty. The explicit inclusion of Shapley uncertainty measures—variance, prediction intervals, or ranking instability—represents a necessary advance in the responsible and interpretable deployment of AI systems.