Structural Causal Model

Each endogenous variable is defined by a deterministic function of its direct causes and an exogenous noise term. Unlike standard regression, these equations represent invariant causal mechanisms, not mere associations.

• Form: X_i = f_i(PA_i, U_i) where PA_i are the parents of X_i and U_i is an unobserved disturbance.
• Invariance: The equation f_i remains stable under interventions on other variables.
• Example: In a supply chain, Delivery_Time = f(Shipping_Mode, Distance, Weather_Noise) captures the physical mechanism, not just a historical correlation.

SCMs explicitly separate the unmodeled background conditions from the variables explained within the system.

• Exogenous (U): Variables determined outside the model, representing unexplained randomness or latent confounders. Their joint distribution P(U) captures all background uncertainty.
• Endogenous (V): Variables whose causal mechanisms are explicitly specified by structural equations. Their values are determined by a combination of other endogenous variables and exogenous noise.
• Supply Chain Example: Supplier_Solvency is often treated as exogenous, while Order_Fulfillment_Rate is endogenous, determined by solvency and other factors.

Every SCM induces a Directed Acyclic Graph (DAG) where nodes are endogenous variables and directed edges represent direct causal dependencies.

• Acyclicity: No variable can be a cause of itself, directly or indirectly. This prohibits feedback loops at the causal level.
• Markovian Assumption: A variable is independent of its non-descendants given its direct parents in the graph, assuming no latent confounders.
• Disruption Analysis: A DAG for a port closure might show edges from Port_Closure to Transit_Delay and from Transit_Delay to Stockout_Risk, enabling root cause tracing.

The do(X=x) operator represents an external intervention that forces a variable to a specific value, breaking its incoming causal arrows. This is the mathematical foundation for answering 'What if?' questions.

• Mechanism: Applying do(X=x) replaces the structural equation for X with the constant x, creating a mutilated graph.
• Distinction: P(Y | X=x) (seeing) is not the same as P(Y | do(X=x)) (doing). The former is confounded; the latter is causal.
• Use Case: To estimate the effect of a forced supplier change (do(Supplier=New)) on lead time, you must sever the influence of the old supplier's performance history.

SCMs are the only framework that fully supports counterfactual queries—reasoning about what would have happened in a specific, past situation had circumstances been different.

• Three-Step Process:
  1. Abduction: Infer the values of exogenous variables (U) given the observed factual evidence.
  2. Action: Apply the do operator to modify the model.
  3. Prediction: Compute the counterfactual outcome using the modified model and inferred U.
• Example: 'Given that a specific shipment was late, would it have been on time if we had used air freight?' This requires updating our belief about unobserved factors (e.g., weather) from the fact of the delay.

A core assumption of SCMs is that each structural equation represents an autonomous mechanism that can be modified independently without altering others.

• Modularity: An intervention on one variable does not change the causal mechanisms of other variables. This allows for localized policy analysis.
• Autonomy: Each mechanism is a stable, independent physical or behavioral law of the system.
• Supply Chain Implication: Changing a warehouse's picking algorithm (do(Picking_Algorithm=New)) should not alter the causal relationship between Order_Volume and Shipping_Time, enabling reliable simulation of localized process changes.