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Explaining Probabilistic Models with Distributional Values
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Introduces distributional values, a generalisation of Shapley-style game-theoretic attributions where payoffs are random variables, giving analytic forms for Bernoulli, Gaussian and categorical games and applying them to image and language model explanations.
Problem
Game-theoretic XAI (e.g. SHAP) requires real-valued payoffs, so probabilistic model outputs (e.g. class distributions) must be reduced to scalars, losing information such as contrastive class transitions and uncertainty.
Approach
Define stochastic cooperative games whose payoffs are random variables sharing noise across coalitions (via reparameterization), and value operators returning distributions of stochastic marginal contributions ("distributional values"), with closed-form expressions for Bernoulli, Gaussian and categorical (Gumbel-argmax) likelihoods.
Key results
- Derives closed-form categorical marginal contribution distributions under Gumbel-argmax reparameterization, computable in O(d) per coalition structure.
- On Iris, feature ordering from standard SV versus categorical SV disagrees for around 80% of training points.
- On XOR game, standard Shapley value gives player 1 zero attribution while distributional values give overall importance 1 for both players.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use distributional Shapley values instead of scalar Shapley values | XOR game · XOR | player 1 Shapley value / overall importance: 0 -> 1 | Helped |
| Use categorical Shapley values instead of scalar Shapley values aggregated across classes | Iris logistic multiclass classifier · Iris training set | feature-ranking discrepancy rate: around 80% | Mixed |
Surprising findings
- Standard Shapley value assigns zero importance to player 1 in the XOR game even though it flips the prediction every time it joins a coalition.
- For a GPT2 gender-stereotype probe, the probability of change for a subject-sex flip is high (0.801) for one sentence pair, suggesting frequent prediction flips.
Limitations
- Computational cost of coalition enumeration/sampling remains exponential; estimation relies on sampling for non-analytic likelihoods.
- Assumes a deterministic model given the noise variable and uses a simple interventional reference (background) strategy for out-of-coalition features.
- Only Bernoulli, Gaussian and categorical likelihoods are treated analytically; general stochastic models are left to future work.
Experimental setup
Datasets: MNIST ImageNet Iris Adult income
Benchmarks: MNIST ImageNet Iris Adult income
Metrics: transition probability overall importance entropy of categorical value feature ranking discrepancy
Baselines: SHAP standard Shapley value (scalar class probability) GP-SHAP
Builds on: Shapley value SHAP semivalues random-order group values Gumbel-argmax reparameterization
Closest papers by meaning
- Problems with Shapley-value-based explanations as feature importance measures (ICML 2020 · similarity 0.89)
- Rethinking Shapley Value for Negative Interactions in Non-convex Games (ICLR 2025 · similarity 0.87)
- Approximating the Shapley Value without Marginal Contributions (AAAI 2024 · similarity 0.87)
- A Unifying Framework to the Analysis of Interaction Methods using Synergy Functions (ICML 2023 · similarity 0.87)
- Unlocking the Game: Estimating Games in Möbius Representation for Explanation and High-Order Interaction Detection (AAAI 2025 · similarity 0.87)
- Axiomatic Aggregations of Abductive Explanations (AAAI 2024 · similarity 0.86)
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