SHAP

The sum of all feature attributions must equal the difference between the model's output for a specific instance and the average model output over the dataset. This axiom guarantees that the prediction is fully and fairly decomposed among the input features.

• Mathematical Form: f(x) = φ₀ + Σᵢ φᵢ
• Genomic Context: For a variant effect predictor, the sum of all nucleotide-level SHAP values exactly accounts for the predicted functional score, leaving no unexplained residual.
• Practical Implication: This property is critical for regulatory compliance, as it provides a complete accounting of a model's decision logic for any single genomic sequence.

A feature that is absent from the input—often represented by a baseline or reference value—must receive an attribution of zero. This ensures that non-existent information does not artificially influence the explanation.

• Implementation: In genomic models, missing features are typically represented by replacing the original nucleotide with a reference token or a neutral embedding vector.
• Edge Case: When analyzing a structural variant, genomic regions that are deleted in the alternate allele are assigned zero importance, preventing spurious attributions to absent sequence.

If a model changes so that a feature's marginal contribution increases or stays the same regardless of other features, the SHAP value for that feature must not decrease. This axiom guarantees that attribution methods are faithful to the underlying model's reliance on features.

• Model Comparison: When comparing two genomic models—one that relies more heavily on a splice site motif and another that does not—the SHAP values for the splice site region will be strictly higher in the first model.
• Sanity Check: This property is tested during Attribution Sanity Checks by randomizing model layers; SHAP values for a randomized model must degrade appropriately, confirming sensitivity to learned parameters.

Two features that contribute identically to every possible subset of other features must receive identical SHAP values. This ensures that the attribution method treats functionally equivalent features without bias.

• Genomic Example: Two perfectly redundant transcription factor binding motifs in a promoter region that always have the same effect on gene expression predictions will be assigned equal importance scores.
• Contrast with Heuristics: Unlike perturbation-based methods that may arbitrarily favor one redundant feature over another due to ordering effects, SHAP's symmetry axiom guarantees deterministic, fair allocation of credit.

When a prediction is the sum of multiple model outputs, the SHAP value for a feature is the sum of its SHAP values from each component model. This property enables the decomposition of complex ensemble predictions.

• Ensemble Genomics: For a gradient-boosted tree ensemble predicting pathogenicity, the SHAP value for a missense variant is the sum of its SHAP values across all individual decision trees.
• Layer-wise Relevance: In deep genomic networks, this axiom allows attributions to be propagated backward through layers, connecting DeepSHAP approximations to the exact Shapley value framework.

Exact SHAP computation requires evaluating all 2^N feature coalitions, which is intractable for high-dimensional genomic sequences. KernelSHAP addresses this by solving a weighted linear regression on a sampled subset of coalitions, prioritizing those with few or many features present.

• Sampling Strategy: For a 1000-base pair sequence, KernelSHAP samples coalitions according to a Shapley kernel weighting scheme, dramatically reducing computation while preserving the theoretical guarantees.
• DeepSHAP Alternative: For deep learning models, DeepSHAP leverages DeepLIFT's backpropagation rules to approximate SHAP values in a single forward and backward pass, achieving linear-time complexity for genomic sequences with millions of nucleotides.

Quantitative measures that evaluate how accurately an attribution map reflects the true decision-making logic of a genomic model. Key metrics include:

• AOPC (Area Over the Perturbation Curve): Measures prediction drop as salient nucleotides are sequentially masked
• ROAR (Retrain After Removal): Iteratively retrains the model after removing top-attributed features to test fidelity
• Infidelity Measure: Quantifies expected error between input perturbations and attribution perturbations These metrics serve as sanity checks for SHAP explanations, ensuring that high-attribution nucleotides are genuinely causal rather than artifacts of the explanation method.