Additive Feature Attribution

The explanation model must match the original model's output for the specific input being explained. If the complex model predicts 0.8 for a given instance, the sum of all feature attributions plus the baseline must equal exactly 0.8. This property ensures the explanation is faithful to the prediction, not an approximation. It is also known as the efficiency property in Shapley value literature, guaranteeing that the total prediction is fully decomposed among the input features without any missing or excess attribution.

A feature that is not present in the original input must receive an attribution of exactly zero. This property ensures that absent or unobserved features do not arbitrarily influence the explanation. In practice, this is handled by defining a baseline value that represents the model's output when no feature information is known. Missingness prevents the explanation model from assigning spurious importance to features that were never provided to the model, maintaining the integrity of the input space.

If a model changes so that a feature's marginal contribution increases or stays the same across all subsets of other features, that feature's attribution must not decrease. This property guarantees that explanations are monotonically aligned with the model's reliance on features. If retraining a model makes a feature more important, the explanation must reflect that increased importance. Consistency ensures that feature attribution methods produce logically coherent explanations that track with model behavior across different model versions or architectures.

The explanation is expressed as a linear sum of individual feature contributions. The final prediction is decomposed into: f(x) = φ₀ + Σ φᵢ, where φ₀ is the baseline expected prediction and each φᵢ is the attribution for feature i. This additive structure is what makes the explanation human-interpretable—each feature pushes the prediction higher or lower from the baseline. Visualizations like waterfall plots and force plots directly leverage this property to show how features combine to produce the final output.

Two features that make identical marginal contributions to every possible coalition of other features must receive identical attributions. This property ensures fair treatment of features that the model uses interchangeably. If swapping two features does not change any prediction, their importance scores must be equal. Symmetry prevents the explanation method from exhibiting arbitrary bias toward certain input dimensions and guarantees that functionally equivalent features are treated identically in the explanation.

A feature that contributes nothing to any prediction—its marginal contribution is zero for all coalitions—must receive an attribution of zero. This is a corollary of the symmetry and efficiency axioms. If a feature never changes the model's output regardless of which other features are present, it is a dummy feature and should be assigned no importance. This property prevents the explanation from manufacturing spurious importance for irrelevant inputs, ensuring that only genuinely influential features receive non-zero attributions.

A simple, interpretable surrogate model used to approximate the behavior of a complex model locally. In additive feature attribution, this is a linear function of binary variables, where each variable indicates the presence or absence of a feature. The explanation model's output is the sum of feature attributions plus a baseline value. This structure ensures that the contribution of each feature is explicitly isolated and directly understandable, transforming an opaque prediction into a transparent, auditable equation.

The property guaranteeing that an explanation model matches the original model's output for a specific input instance. If f(x) is the original model's prediction and g(x') is the explanation model, local accuracy requires f(x) = g(x'). This ensures the explanation is faithful to the prediction being explained, not an approximation of the model's global behavior. It is a fundamental constraint that prevents explanations from drifting away from the actual decision they are meant to illuminate.

A SHAP property requiring that features not present in the original input are assigned an attribution of zero. If a feature is missing from an instance, its corresponding binary variable x' is set to 0, and its Shapley value must be exactly 0. This prevents the explanation from attributing influence to features that did not contribute to the prediction. Missingness ensures that the explanation is sparse and honest, only crediting features that were actually observed and processed by the model.

A Shapley property stating that if a model changes so a feature's marginal contribution increases or stays the same regardless of other features, its SHAP value should not decrease. Formally, if f_x(S ∪ {i}) - f_x(S) ≥ f'_x(S ∪ {i}) - f'_x(S) for all subsets S, then the attribution for feature i in model f must be at least as large as in model f'. This guarantees that explanations monotonically track feature importance across model updates, preventing counterintuitive reversals in attribution rankings.

A Shapley axiom ensuring that the sum of all feature attributions exactly equals the difference between the model's prediction and the baseline value. Mathematically, ∑ φ_i = f(x) - E[f(X)]. This property guarantees that the explanation is complete and additive—no influence is lost or double-counted. The total prediction is fully decomposed into individual feature contributions, enabling precise accounting of how each input pushes the output away from the expected value.

The expected model output across the background dataset, representing the prediction when no feature information is known. It serves as the starting point for additive decomposition. In SHAP, the baseline is typically E[f(X)], the average prediction over the background distribution. All feature attributions are measured relative to this reference point. Choosing an appropriate baseline is critical: a poorly chosen baseline can distort explanations by anchoring them to an unrepresentative or unrealistic reference value.