Causal Shapley

The fundamental distinction from standard Shapley values is the use of interventional distributions rather than conditional distributions. Standard SHAP asks, 'What is the model's prediction when we know feature X?' Causal Shapley asks, 'What is the outcome when we set feature X?' This shift from P(Y | X=x) to P(Y | do(X=x)) ensures the attribution reflects causal influence, not just statistical association, by blocking backdoor paths through other features.

Attribution respects the partial ordering defined by the causal graph (DAG). A feature can only be credited for effects flowing through its descendants. When evaluating a coalition's value, features are 'intervened on' according to this order, preventing illogical attributions where an effect is credited to a variable that occurs later in the causal chain. This property ensures the explanation aligns with the known or assumed causal mechanics of the system.

Causal Shapley values provide an additive decomposition of the total causal effect of moving all features from a baseline state to their current values. For a given outcome difference Y(do(X=x)) - Y(do(X=x')), the sum of the Causal Shapley values for all features equals this total effect. This property guarantees that the attribution is a complete and fair accounting of the measured causal change.

The method satisfies the four classic Shapley axioms, reinterpreted for causal settings:

• Efficiency: The attributions sum to the total causal effect.
• Symmetry: Two features that have identical causal effects receive equal attribution.
• Dummy: A feature with no causal effect on the outcome receives zero attribution.
• Additivity: For a combined effect, the attribution is the sum of attributions from individual effects. These axioms provide a game-theoretically fair and unique solution to the attribution problem.

The framework correctly attributes effects through direct and indirect pathways. By using the do-operator, it accounts for confounding: the effect attributed to a feature is not inflated by backdoor associations. It can also decompose a feature's total effect into portions mediated through other variables versus direct effects, providing nuanced insight into the causal mechanism, which is crucial for interpretable and robust model explanations.

Calculation requires estimating many interventional distributions (P(Y | do(S)) for all feature subsets S), which is computationally intensive and requires causal identifiability. The effect for a coalition must be identifiable from the available data and causal graph, often relying on assumptions like no unmeasured confounding within the subset. This makes Causal Shapley values both more principled and more demanding than their associational counterpart.

The Shapley value is a solution concept from cooperative game theory that provides a unique, axiomatically fair method for distributing the total payoff of a coalition game among its players. It is defined as the average marginal contribution of a player across all possible orderings of players joining the coalition.

• Core Concept: Fair attribution based on average marginal contribution.
• Key Axioms: Efficiency, Symmetry, Dummy Player, and Additivity.
• Application in ML: The foundation for SHAP (SHapley Additive exPlanations), used for feature attribution in predictive models.

Causal inference is the process of drawing conclusions about cause-and-effect relationships from data, moving beyond statistical associations to determine the impact of an intervention or treatment on an outcome. It relies on formal frameworks like Structural Causal Models (SCMs) and the do-calculus.

• Goal: Answer "what if" questions about interventions.
• Key Tools: Randomized controlled trials, observational studies with adjustment (e.g., via backdoor criterion), instrumental variables.
• Contrast with Prediction: Focuses on understanding system dynamics under change, not just forecasting.

A Structural Causal Model (SCM) is a formal mathematical framework that represents causal relationships between variables using a system of structural equations, typically visualized as a causal graph (a Directed Acyclic Graph). It defines how each variable is generated from its direct causes and independent noise.

• Components: A set of variables, a set of error terms, and a set of functions assigning each variable a value based on its parents.
• Enables: Reasoning about interventions (do-operator) and counterfactuals.
• Role in Causal Shapley: Provides the underlying causal graph and structural equations necessary to compute the value of a feature under different "worlds" (interventional distributions).

A counterfactual is a statement about what would have happened to an outcome if a cause had been different. It represents the highest level of reasoning on Pearl's 'ladder of causation' and answers retrospective 'what if' questions for specific instances.

• Example: "Would this patient have survived if they had not received the drug?"
• Computation: Requires a fully specified SCM to account for background conditions.
• Connection to Causal Shapley: Causal Shapley values can be viewed as a weighted average of counterfactual contributions of a feature across all possible feature subsets, moving beyond purely associational Shapley values.

Do-calculus is a set of three inference rules developed by Judea Pearl that allows one to compute the effects of interventions from observational data, provided a causal graph is known. It transforms expressions containing the do-operator (e.g., P(Y|do(X))) into observational probabilities.

• Purpose: To identify and estimate causal effects from non-experimental data.
• Prerequisite: A valid causal graph representing the data-generating process.
• Role in Causal Shapley: Provides the formal machinery to compute the interventional distributions required when evaluating a feature's contribution within a coalition (subset of variables). It replaces the observational conditional distributions used in standard Shapley with interventional ones.

The Average Treatment Effect (ATE) is the average causal effect of a treatment or intervention across an entire population. It is calculated as the expected difference in outcomes between the treated and untreated states for a randomly selected individual: ATE = E[Y|do(T=1)] - E[Y|do(T=0)].

• Population-Level: Measures the effect for the group, not for a specific individual.
• Estimation Methods: Includes randomized experiments, propensity score matching, and double machine learning.
• Contrast with Causal Shapley: While ATE attributes the total effect to a single treatment variable, Causal Shapley decomposes a complex outcome (which may be a function of multiple treatments/features) and attributes portions of it to each contributing variable in a causally-grounded, fair manner.