Conditional Expectation

Conditional expectation estimates the value of a missing feature by averaging over the observed values of other features it depends on. In the SHAP context, when a feature is 'absent' from a coalition, its value is not set to zero or a global mean—it is imputed based on the known, present features. This preserves the joint distribution of the data.

• Formula: E[f(x) | x_S], where x_S are the known features
• Key Distinction: Differs from marginal expectation, which ignores dependencies
• Result: Model evaluations occur only on realistic data points

Computing the exact conditional expectation is intractable for high-dimensional data. In practice, it is approximated using a representative background dataset. For a given instance with known features x_S, the algorithm finds similar rows in the background data and averages their outcomes.

• K-Nearest Neighbors: A common non-parametric estimator that weights nearby background samples
• Assumption: The background dataset must be a faithful sample of the training distribution
• Sensitivity: Sparse regions of the feature space yield high-variance estimates

The primary motivation for conditional expectation is to avoid evaluating the model on off-manifold data points. If height and weight are correlated, an interventional approach might ask the model to predict for a 7-foot-tall, 90-pound person. Conditional expectation ensures that when height is known, weight is imputed realistically.

• Avoids extrapolation: Keeps model inputs within the training distribution
• Captures redundancy: Correctly splits credit among correlated features
• Real-world example: In credit scoring, income and debt-to-income ratio are linked; conditional expectation respects this link

Estimating E[f(x) | x_S] for every possible coalition S is the bottleneck of Observational SHAP. Unlike TreeSHAP's polynomial-time exact computation for interventional SHAP, conditional expectation requires solving a distinct density estimation problem for each subset of features.

• Complexity: Grows exponentially with the number of features in the worst case
• Approximations: TreeSHAP can be adapted for observational SHAP by modeling the conditional distribution within tree structures
• Practical impact: Often necessitates sampling-based approximations for large feature sets

Conditional expectation directly satisfies the missingness axiom in SHAP. This axiom requires that a feature not present in the original input (a 'missing' feature) receives zero attribution. By imputing missing features based on what is known, the baseline prediction already accounts for their expected influence.

• Axiom compliance: Ensures x_i = 0 implies φ_i = 0
• Mechanism: The contribution of a missing feature is absorbed into the baseline through the conditional expectation
• Contrast: Simple mean imputation would violate this property by assigning non-zero attribution to absent features

The SHAP formulation that directly applies conditional expectation to preserve feature correlations. When a feature is 'missing' from a coalition, its value is imputed by conditioning on the known features present in that coalition.

• Reflects the model's behavior on the natural data manifold
• Avoids evaluating the model on unrealistic, off-manifold data points
• Computationally more expensive due to the need to estimate conditional densities
• Preferred when explaining correlations is critical to the domain understanding

A causal interpretation of SHAP that breaks feature correlations by sampling from the marginal distribution rather than conditioning. This approach answers 'what if' questions by simulating physical interventions on the system.

• Breaks correlations between the feature of interest and other features
• Reflects the model's behavior under external manipulation
• Computationally simpler, using random sampling from the background dataset
• Preferred for causal reasoning and fairness audits

A representative sample of data used to compute the expected model output and to impute missing features during SHAP value estimation. For conditional expectation, this dataset defines the joint distribution from which conditional densities are estimated.

• Serves as the reference population for explanations
• Must be representative of the data the model encounters in production
• Size impacts the accuracy of conditional probability estimates
• The choice between a single background sample and a full distribution affects the explanation's stability

The fundamental building block of Shapley value computation. It is the difference in the model's prediction when a specific feature is added to a subset of other features. Conditional expectation determines how the model's output is evaluated for coalitions where the feature is absent.

• Formula: v(S ∪ {i}) - v(S) where v is the value function
• The value function v(S) integrates out missing features using conditional expectation
• Averaged over all possible coalitions to compute the Shapley value
• The choice of expectation method fundamentally alters the marginal contribution's meaning

The simplifying assumption that input features are uncorrelated, which reduces computational complexity but may produce unrealistic model evaluations. Under this assumption, conditional expectation simplifies to the marginal expectation.

• Allows E[f(x) | x_S] to be approximated by E[f(x_S, X_{not S})]
• Reduces the need for complex conditional density estimation
• Can generate off-manifold data points that the model has never seen
• Violated in most real-world datasets, leading to potential explanation artifacts

The mathematical field studying how groups of players form coalitions and distribute payoffs, providing the theoretical foundation for Shapley values. Conditional expectation defines the 'worth' of a feature coalition by computing the model's expected output given only the features in that coalition.

• Each feature is a player; the model prediction is the payout
• The characteristic function v(S) is defined via conditional expectation: E[f(X) | X_S = x_S]
• Shapley values are the unique solution satisfying efficiency, symmetry, dummy, and additivity axioms
• The axiomatic foundation ensures fair and consistent feature attribution