Do-Calculus

Rule Statement: P(y | do(x), z, w) = P(y | do(x), w) if (Y ⊥⊥ Z | X, W) in the graph G_\overline{X}.

This rule permits adding or removing conditioning variables from a probability expression involving a do-operator, provided those variables are conditionally independent of the outcome Y in the manipulated graph. The manipulated graph, denoted G_\overline{X}, is the original causal graph with all incoming arrows to the intervened variable X removed.

• Use Case: Simplifying an estimand by removing irrelevant observed variables that do not provide additional information about Y once X is intervened upon and W is conditioned on.
• Key Insight: Observations (Z) can be ignored if they are d-separated from Y in the post-intervention graph. This rule formalizes when 'controlling for' a variable is unnecessary.

Rule Statement: P(y | do(x), do(z), w) = P(y | do(x), z, w) if (Y ⊥⊥ Z | X, W) in the graph G_{\overline{X}, \underline{Z}}.

This powerful rule allows an intervention do(z) to be replaced by the observation of Z, or vice-versa, under specific graphical conditions. The graph G_{\overline{X}, \underline{Z}} is the original graph with arrows into X removed and arrows out of Z removed.

• Use Case: Eliminating an inner do-operator to express a quantity with fewer interventions, moving it closer to a purely observational form.
• Key Insight: An intervention can be treated as a passive observation if, in the modified graph where Z's causes are ignored, Z is d-separated from Y. This often applies when Z is not a mediator on a path between X and Y after the initial intervention.

Rule Statement: P(y | do(x), do(z), w) = P(y | do(x), w) if (Y ⊥⊥ Z | X, W) in the graph G_{\overline{X}, \overline{Z(W)}}.

This rule permits adding or removing an irrelevant intervention. The graph G_{\overline{X}, \overline{Z(W)}} removes arrows into X and into any node in Z that is not an ancestor of W in the original graph.

• Use Case: Removing a do-operator that has no effect on the outcome Y, given the other interventions and observations. This simplifies the causal query.
• Key Insight: An intervention can be ignored if it does not influence the outcome variable in the context of the other performed interventions. It formalizes the intuition of an irrelevant action within the causal system.

The do-operator, denoted do(X=x), is the mathematical representation of an intervention that sets a variable X to a specific value x, overriding its natural causal mechanisms. It answers interventional queries like 'What would the rate of disease be if we forced everyone to take the drug?'

• Graphical Representation: An intervention do(X) is represented by modifying the original causal directed acyclic graph (DAG). All incoming edges to X are deleted, as its value is now set externally, breaking its dependence on its usual causes.
• Contrast with Conditioning: P(Y | X=x) describes association within observed data. P(Y | do(X=x)) describes the causal effect of actively changing X. The backdoor criterion and front-door criterion are graphical tests to identify sets of variables that, when adjusted for, allow P(Y | do(X)) to be equated to an observational formula.

A central result in causal inference is that do-calculus is complete. This means that if a causal effect is identifiable from observational data given the causal graph, there exists a sequence of do-calculus rule applications that will transform P(y | do(x)) into an expression containing only observational probabilities (no do-operators).

• Identifiability: A causal query is identifiable if it can be uniquely computed from the observed probability distribution P(V) of the variables V in the graph. Non-identifiability arises from unobserved confounders creating ambiguous causal pathways.
• Algorithmic Application: The ID algorithm automates the application of do-calculus rules to determine identifiability and derive the correct estimand. It systematically searches for a valid sequence of transformations.
• Implication: Do-calculus provides a sound and complete 'toolbox' for solving causal identification problems, moving beyond heuristic adjustments.

Consider a classic confounded model: a treatment X, outcome Y, and an observed confounder Z. The graph is Z → X → Y and Z → Y. We want P(y | do(x)).

1. Goal: Transform P(y | do(x)) to an observational formula.
2. Apply Rule 2: We ask if we can change do(x) to see(x). Check condition: Is (Y ⊥⊥ X | Z) in G_{\underline{X}}? (Graph with arrows out of X removed). In this graph, the path X ← Z → Y is active, so they are not independent. Rule 2 fails.
3. Apply Rule 3: Can we delete an action? Not applicable.
4. Alternative Strategy - Conditioning: Start with P(y | do(x)) = Σ_z P(y, z | do(x)) = Σ_z P(y | do(x), z) P(z | do(x)).
5. Apply Rule 2 to P(y | do(x), z): In G_{\underline{X}}, is (Y ⊥⊥ X | Z)? No, as before.
6. Apply Rule 3 to P(z | do(x)): In G_{\overline{X}}, is (Z ⊥⊥ X)? Yes, because the edge from Z to X is removed. So by Rule 3, P(z | do(x)) = P(z).
7. Apply Rule 2 to P(y | do(x), z) in a different graph: Consider G_{\overline{X}}. In this graph, with arrows into X removed, the only path from X to Y is X → Y, which is blocked by conditioning on nothing. However, we are conditioning on Z. In G_{\overline{X}}, Z is a non-collider on the path X → Y ← Z? No, that path doesn't exist. Actually, in G_{\overline{X}}, X and Y are d-separated given Z? Let's check: The path X → Y is blocked by conditioning on Y's child? Wait, Z is a parent of Y. The path is X → Y ← Z. This is a collider at Y? No, Y is not a collider here because both arrows point to Y. This is a 'chain' where Z influences Y. Conditioning on Z opens the path? No, conditioning on a non-collider does not open a path. In G_{\overline{X}}, the path X → Y is direct and unblocked. Conditioning on Z, a parent of Y, does not block the direct path. So X and Y are not independent given Z in G_{\overline{X}}. Therefore, Rule 2 does not apply directly to P(y | do(x), z).

The correct, simpler derivation leverages the backdoor criterion: Z satisfies it for (X, Y). The do-calculus derivation is: P(y|do(x)) = Σ_z P(y|do(x), z)P(z|do(x)) [Law of Total Probability]. By Rule 3, P(z|do(x)) = P(z) as shown. Now, in G_{\overline{X}}, the only path from X to Y is the direct edge. Is (Y ⊥⊥ X | Z) in G_{\overline{X}}? In G_{\overline{X}}, the path X → Y is still present. However, if we consider the mutually exclusive graph manipulation for Rule 2, which is G_{\overline{X}, \underline{Z}}, we remove arrows into X and arrows out of Z. In this graph, Z has no arrows out, so the path X → Y ← Z is severed. Therefore, Y is d-separated from X given Z in G_{\overline{X}, \underline{Z}}. Applying Rule 2: P(y | do(x), z) = P(y | x, z).

Final Result: P(y | do(x)) = Σ_z P(y | x, z) P(z). This is the backdoor adjustment formula, derived rigorously via do-calculus.

A Structural Causal Model is the formal mathematical framework upon which do-calculus operates. It consists of:

• Structural Equations: Functions that assign values to variables based on their direct causes.
• Causal Graph (DAG): A visual representation of the causal relationships, where nodes are variables and edges indicate direct causation.
• Exogenous Variables: Unmodeled background factors, represented as noise. Do-calculus provides the rules for querying this model to answer interventional questions like P(Y | do(X)).

Interventional Inference is the core task enabled by do-calculus: predicting the effects of actions or forced changes. It answers 'what if' questions, contrasting with purely associative (observational) reasoning.

Key Distinction:

• Observational: P(Y | X = x) - Seeing that X is x.
• Interventional: P(Y | do(X = x)) - Making X be x. Do-calculus provides the mathematical machinery to compute the latter from the former, given a causal graph, even in the presence of confounding.

A Causal Graph or Directed Acyclic Graph (DAG) is the indispensable input to do-calculus. It encodes domain knowledge about cause-and-effect relationships.

Critical Elements for Do-Calculus:

• Nodes: Represent random variables.
• Directed Edges: Represent direct causal influence (X → Y means X causes Y).
• Acyclicity: No variable can be its own ancestor (no feedback loops in the model).
• Implicit Assumptions: Absence of an edge is a strong assumption of no direct causal effect. The three rules of do-calculus are defined in terms of graphical relationships (d-separation) within this DAG.

The Backdoor Criterion is a graphical pre-calculus method for identifying a set of variables to adjust for in order to estimate a causal effect. It is often used before applying do-calculus to simplify the problem.

A set of variables Z satisfies the backdoor criterion for (X, Y) if:

1. Z blocks every backdoor path (non-causal path) between X and Y.
2. No node in Z is a descendant of X. If such a Z exists, the causal effect is identifiable via adjustment: P(Y | do(X)) = Σ_z P(Y | X, Z=z) P(Z=z). Do-calculus can derive this formula and handle cases where simple adjustment fails.

The Frontdoor Criterion provides an identification strategy for causal effects when a confounder is unobserved and the backdoor criterion cannot be applied. Do-calculus can formally validate and derive the frontdoor adjustment formula.

A set of variables M satisfies the frontdoor criterion for (X, Y) if:

1. M mediates all directed paths from X to Y.
2. There is no unblocked backdoor path from X to M.
3. X blocks all backdoor paths from M to Y. The causal effect is then: P(Y | do(X)) = Σ_m P(M=m | X) Σ_x' P(Y | X=x', M=m) P(X=x'). Do-calculus proves this equivalence from the graph alone.

Counterfactual Reasoning operates at the third and deepest level of Pearl's Causal Hierarchy (Association → Intervention → Counterfactuals). It answers retrospective, 'what would have been' questions about specific instances.

Example: 'Would patient Y have survived if they had received the drug, given that they did not?'

Relationship to Do-Calculus:

• Do-calculus deals with population-level interventions (P(Y | do(X))).
• Counterfactuals require additional machinery (like Structural Equation Models and the concept of external intervention) to model individual-level potential outcomes. Do-calculus is a foundational component for identifying the estimands needed for counterfactual analysis.