Causal Graph

A causal graph is formally a DAG, meaning all edges have a direction and no path loops back to its origin. The direction of an edge X → Y encodes the assumption that X is a direct cause of Y. Acyclicity ensures no variable can cause itself, preventing logical paradoxes. This structure allows the graph to be factorized into a joint probability distribution where each node is conditioned only on its direct parents, forming the basis for the Causal Markov Condition.

Every node represents a random variable in the system—this can be an observed feature, a latent confounder, or a treatment. A directed edge X → Y asserts that X has a direct causal effect on Y that is not fully mediated by any other variable in the graph. The absence of an edge is a strong claim of no direct effect. This sparse encoding forces modelers to explicitly state their causal assumptions, making the graph a transparent scientific model rather than a black-box correlation matrix.

This property states that a variable is independent of all its non-descendants given its direct parents in the graph. Formally: X ⊥ NonDescendants(X) | Parents(X). This allows the joint distribution to be factorized as P(V) = ∏ P(v | parents(v)). The Markov condition is the critical bridge between the graphical structure and the statistical data, enabling the decomposition of a complex high-dimensional distribution into modular, estimable components.

d-separation is the graphical criterion for reading all conditional independencies implied by the Markov condition. Two sets of nodes are d-separated by a third set if every path between them is blocked. A path is blocked by:

• A chain X → M → Y or fork X ← M → Y where M is conditioned on.
• A collider X → C ← Y where neither C nor its descendants are conditioned on. d-separation allows analysts to test their causal graph against data by verifying that implied independencies hold empirically.

The faithfulness assumption is the converse of the Markov condition: all conditional independencies observed in the data must be entailed by the graph's structure via d-separation. Violations occur when causal pathways cancel each other out perfectly—for example, two paths with equal and opposite effects. Faithfulness is assumed by most causal discovery algorithms. Without it, the true causal graph may be statistically indistinguishable from a structurally different graph, complicating structure learning from observational data.

An intervention is an external manipulation that sets a variable to a specific value, breaking its dependence on its natural causes. Graphically, this is represented by deleting all incoming edges to the intervened node. The do-operator P(Y | do(X=x)) denotes the post-intervention distribution, which is fundamentally different from conditioning P(Y | X=x). This distinction is the core of causal reasoning: conditioning observes a passive world, while the do-operator simulates an active change, enabling counterfactual queries.

The formal mathematical framework that defines a causal graph. An SCM specifies a set of endogenous variables (nodes), exogenous variables (unexplained noise), and structural equations (edges) that represent the data-generating process.

• Equations: Each child node is a deterministic function of its parents and an independent noise term: X_i = f_i(PA_i, U_i).
• Intervention: The do(X=x) operator surgically removes a variable from its parents, breaking incoming edges.
• Counterfactual Computation: The three-step process of abduction (inferring noise), action (intervening), and prediction (recomputing) relies entirely on the SCM.

A set of three inference rules developed by Judea Pearl that determine when a causal effect—expressed as an interventional distribution P(Y|do(X))—can be estimated from purely observational data and the structure of the causal graph.

• Rule 1 (Insertion/Deletion of Observations): Allows adding or removing a variable from the conditioning set if Y and Z are d-separated by X in a modified graph.
• Rule 2 (Action/Observation Exchange): Allows replacing an intervention with a conditioning event if they share the same effect via backdoor paths.
• Rule 3 (Insertion/Deletion of Actions): Allows removing an intervention entirely if the effect is independent of the intervention in the relevant subgraph.
• Completeness: If a causal effect is identifiable, it can be derived by repeated application of these three rules.

A rigorous definition of individual fairness grounded in causal graphs. A predictor Ŷ is counterfactually fair if its output for an individual is identical in the actual world and a counterfactual world where a sensitive attribute A (e.g., race, gender) was different.

• Causal Dependence: The definition only penalizes unfairness transmitted through causal paths from the sensitive attribute to the outcome.
• Non-Causal Correlation: Correlations due to common causes or random sampling are not considered unfair under this strict criterion.
• Graph Requirement: Computing this metric requires a fully specified causal graph that includes the sensitive attribute and all its descendants.

A hard rule encoded into a counterfactual search algorithm that prevents the generation of explanations that violate real-world causal laws or immutable characteristics. This transforms a generic counterfactual into an actionable recourse.

• Causal Constraints: Enforced by the causal graph's directed edges. Changing a parent node propagates changes to its descendants, preventing physically impossible counterfactuals.
• Immutability: Variables like age or place of birth are clamped and cannot be modified by the optimization algorithm.
• Action Set: The formal specification of which features can be changed, by how much, and in which direction, defining the feasible region for the counterfactual search.

The three-step computational process of answering 'what if' questions using a fully specified SCM. This is the algorithmic engine that powers all counterfactual explanations.

• Step 1: Abduction: Observe the factual outcome and use the structural equations to infer the posterior distribution of the unobserved exogenous noise variables U.
• Step 2: Action: Apply the do() operator to the causal graph, surgically setting a variable to a counterfactual value and severing all incoming edges.
• Step 3: Prediction: Recompute the values of the outcome variable in the modified graph using the inferred noise distribution from Step 1.
• Deterministic vs. Probabilistic: With deterministic structural equations, this yields a point estimate; with probabilistic equations, it yields a distribution.

The property that a counterfactual recommendation remains valid even after the underlying predictive model is retrained or slightly updated. A lack of robustness leads to recourse invalidation, where a user implements the recommended changes but the new model still denies them.

• Model Shift: Retraining on new data can shift the decision boundary, invalidating previously generated counterfactuals.
• Robust Algorithms: Methods like Causal Recourse that constrain changes to follow the causal graph's mechanisms are inherently more robust than purely distance-based methods.
• Evaluation: Measured by the percentage of counterfactuals that remain valid after a model update, a critical metric for deploying recourse systems in production.