Feature Independence Assumption

The SHAP formulation that preserves feature correlations by conditioning on observed values rather than assuming independence. It computes the conditional expectation of the model output given a subset of features, reflecting how the model behaves on the natural data manifold. This approach avoids evaluating the model on unrealistic data points but breaks the interventional interpretation of Shapley values, potentially attributing importance to features the model does not causally use.

A causal interpretation of SHAP that explicitly breaks feature correlations by sampling from the marginal distribution. It answers: 'How does the model output change if we intervene and set a feature to a specific value, independent of others?' This formulation satisfies the Shapley axioms perfectly but may evaluate the model on out-of-distribution data points when features are correlated, producing explanations that reflect the model's structure rather than real-world data patterns.

An extension of SHAP that incorporates a causal directed acyclic graph (DAG) to compute feature attributions respecting the underlying causal structure. Instead of assuming independence or conditioning on all observed features, Causal SHAP uses do-calculus to simulate interventions along causal paths. This resolves the tension between Observational and Interventional SHAP by attributing importance only to features that are causal ancestors of the prediction, blocking spurious correlations from confounders.

The statistical method used in Observational SHAP to estimate missing feature values by conditioning on known features. When a feature is absent from a coalition, its value is imputed by computing the expected value given the present features: E[f(x) | x_S]. This preserves the joint distribution but is computationally expensive, often requiring separate models to estimate the conditional distribution for each subset of features.

A representative sample of data used to compute the expected model output and to impute missing features during SHAP value estimation. The choice of background dataset directly impacts the baseline value and the resulting attributions. When features are correlated, using the full background dataset for marginal sampling (Interventional SHAP) can create unrealistic feature combinations, while conditioning on the background (Observational SHAP) preserves realism but increases computational cost.

The process of approximating exact Shapley values using sampling techniques when exhaustive computation over the power set (2^M coalitions) is infeasible. The Feature Independence Assumption simplifies estimation by allowing features to be sampled independently from their marginal distributions. When this assumption is violated, estimation becomes more complex: conditional sampling requires modeling the joint distribution, and variance reduction techniques like antithetic sampling become critical to maintain convergence with correlated features.