SHAP (SHapley Additive exPlanations)

The sum of all feature attributions for a specific prediction exactly equals the difference between the model's output for that instance and the average model output over the dataset. This property, also known as efficiency, guarantees that the explanation is a complete decomposition of the prediction.

• Equation: f(x) = φ₀ + Σᵢ φᵢ, where φ₀ is the base value (expected prediction) and φᵢ are the SHAP values.
• Fraud Context: For a transaction scored at 0.85 (high risk) against a baseline of 0.10, the SHAP values for 'transaction amount', 'time of day', and 'device fingerprint' will sum to exactly +0.75, leaving no unexplained variance.

A feature that is not present in the input—either because it is genuinely missing or was intentionally excluded—is assigned an attribution of exactly zero. This prevents the model from inventing importance for absent data.

• Mechanism: SHAP handles missing features by integrating them out using the background dataset's distribution, ensuring they contribute nothing to the explanation.
• Practical Impact: If a fraud model does not receive a 'billing address match' signal for a transaction, that feature's SHAP value is strictly zero, preventing spurious attributions that could mislead investigators.

If a model is changed so that a feature's contribution increases or stays the same regardless of other features, the SHAP value for that feature will never decrease. This monotonicity property ensures that explanations track logically with model updates.

• Formal Definition: If f'(S ∪ {i}) - f'(S) ≥ f(S ∪ {i}) - f(S) for all subsets S, then φᵢ(f') ≥ φᵢ(f).
• Fraud Model Evolution: When retraining a model to rely more heavily on 'login geolocation velocity', the SHAP value for that feature will only increase or stay flat—it will never contradict the model's increased reliance by showing a lower importance.

Two features that contribute identically to every possible subset of features receive identical SHAP values. This ensures that the attribution method treats functionally equivalent features with absolute fairness.

• Implication: If 'card present flag' and 'chip read success' always have the same marginal effect on the fraud score when added to any coalition of other features, their SHAP values will be exactly equal.
• Audit Utility: This property is critical for regulatory compliance, as it guarantees that no feature is arbitrarily favored over another that provides the same predictive signal, supporting defensible adverse action reason codes.

The SHAP value for a feature in an ensemble model is the weighted average of its SHAP values from each constituent model. This allows explanations to be decomposed across complex model architectures.

• Ensemble Decomposition: For a fraud detection ensemble combining a gradient boosting model and a neural network, the global SHAP value for 'transaction velocity' is the weighted sum of its SHAP values from each sub-model.
• Layer-wise Analysis: This property extends to explaining multi-stage pipelines, where the final attribution can be traced back through feature transformations, enabling granular debugging of feature engineering steps.

SHAP values are the only additive feature attribution method that simultaneously satisfies local accuracy, missingness, and consistency. This result, proven by Lundberg and Lee, unifies several prior explanation methods under a single optimal framework.

• Unified Methods: KernelSHAP, TreeSHAP, and DeepSHAP are all specific implementations that compute Shapley values efficiently for different model types, all converging to the same unique solution.
• Regulatory Gold Standard: For generating adverse action reason codes under regulations like the Fair Credit Reporting Act (FCRA), the uniqueness property provides a legally defensible, mathematically singular explanation that cannot be contradicted by another valid method.