Additive Feature Attribution

The explanation model must perfectly match the original model's output for the specific input being explained. If f(x) is the original prediction and g(x') is the additive explanation, then f(x) = g(x') = φ_0 + Σ φ_i.

• Property: The sum of all feature attributions plus a baseline equals the model's exact prediction.
• Mechanism: The baseline φ_0 represents the model's output when all features are 'missing' or set to a reference value.
• Constraint: This forces the explanation to be a faithful, zero-error decomposition of the single prediction, not an approximation.

If a feature is missing from the original input—meaning its value is set to a baseline or reference point—its attribution must be exactly zero.

• Property: x'_i = 0 implies φ_i = 0.
• Purpose: Ensures that features which do not contribute to the prediction are not arbitrarily assigned importance.
• Implementation: In SHAP, missingness is handled by integrating over the feature's distribution; in LIME, it is enforced by the interpretable representation where a '0' indicates absence.
• Contrast: This is a critical distinction from simple gradient-based methods, which can assign non-zero importance to features with zero value.

If a model changes so that a feature's marginal contribution increases or stays the same regardless of other features, that feature's attribution must not decrease.

• Property: If f'_x(S ∪ i) - f'_x(S) ≥ f_x(S ∪ i) - f_x(S) for all subsets S, then φ_i(f', x) ≥ φ_i(f, x).
• Significance: This axiom guarantees that a feature's importance score is logically monotonic with its actual impact on the model.
• Uniqueness: Combined with local accuracy and missingness, consistency forces a single, unique solution—the Shapley values.

The three core properties—local accuracy, missingness, and consistency—collectively prove that Shapley values are the only possible additive feature attribution method that satisfies all axioms.

• Game Theory Origin: Shapley values were originally derived to fairly distribute a coalition's payout among players based on their marginal contributions.
• ML Translation: In machine learning, 'players' become input features, and the 'payout' is the model's prediction relative to a baseline.
• SHAP Framework: SHAP (SHapley Additive exPlanations) is the direct implementation of this proof, unifying LIME, DeepLIFT, and other methods under a single theoretical umbrella.

Additive explanations decompose a prediction into a sum of independent feature contributions, but real models often rely on complex feature interactions.

• Limitation: An additive explanation cannot directly represent interaction effects where the combined impact of two features differs from the sum of their individual impacts.
• Workaround: SHAP extends the framework with Shapley interaction values, which decompose the prediction into a matrix of main effects and pairwise interaction terms.
• Trade-off: While interaction values capture more model complexity, they sacrifice the simplicity and human-interpretability of a purely additive bar chart.

The choice of the baseline or reference value is a critical, non-trivial design decision that anchors the entire additive explanation.

• Role: The baseline represents the 'absence' of information and determines the starting point φ_0 from which feature contributions are measured.
• Common Choices: An all-black image for vision models, a zero-vector or [MASK] token for text, or the dataset mean for tabular data.
• Impact: A poorly chosen baseline can produce misleading attributions. For example, using a zero pixel value may conflate 'missing' with 'black'.
• Best Practice: The baseline should be an in-distribution, neutral input that represents a true absence of signal for the specific domain.

A solution concept from cooperative game theory that determines the marginal contribution of each player (feature) averaged over all possible coalitions. In ML explainability, the 'game' is the prediction task, and the 'payout' is the difference between the model's output and the average prediction.

• Satisfies efficiency: feature contributions sum to the prediction difference from baseline
• Satisfies symmetry: features with identical marginal contributions receive identical values
• Satisfies dummy: a feature that never contributes receives zero attribution

The reference point against which feature contributions are measured in additive attribution methods. The baseline represents the expected model output when no feature information is present, often computed as the average prediction over the training dataset.

• For tabular data: the mean or median of each feature column
• For images: a black image, blurred version, or zero-valued tensor
• For text: an empty string or [MASK] token
• The choice of baseline critically impacts explanation interpretation and fairness

The defining mathematical constraint of this explanation class: the sum of all feature attributions plus the baseline value must exactly equal the model's prediction for that instance. This ensures completeness and prevents unexplained variance.

• Enables linear decomposition of complex non-linear predictions
• Allows attributions to be interpreted as additive contributions in the original output units
• Contrasts with gradient-based methods that measure sensitivity rather than contribution

A model-agnostic implementation of SHAP that extends the LIME framework by using a Shapley-weighted kernel for local surrogate model training. KernelSHAP solves the additive attribution problem by weighting perturbed samples according to their Shapley kernel weight rather than an arbitrary distance metric.

• Provides theoretically grounded local explanations for any black-box model
• Computationally expensive: requires 2^M model evaluations for M features in exact form
• Practical implementations use sampling approximations to reduce computation

A fast, exact algorithm for computing SHAP values specifically for tree-based models like XGBoost, LightGBM, and random forests. TreeSHAP exploits the internal structure of decision trees to compute attributions in polynomial time rather than exponential.

• Reduces complexity from O(2^M) to O(TLD^2) where T=trees, L=leaves, D=depth
• Computes exact SHAP values without sampling or approximation
• Supports feature interactions by decomposing attributions into main effects and interaction terms