Pearl's Causal Calculus

About Causal Inference

Judea Pearl's causal inference framework distinguishes between mere correlation and genuine causation using directed acyclic graphs (DAGs) and the do-operator. This is fundamental to understanding when we can legitimately infer cause from effect.

Causal Graphs (DAGs)

A causal graph represents variables as nodes and direct causal relationships as directed edges. The structure encodes which variables causally influence others:

• Fork (X ← Z → Y): Z is a common cause (confounder) of both X and Y. Observing Z blocks the spurious correlation.
• Chain (X → Z → Y): Z is a mediator. The effect of X on Y flows through Z.
• Collider (X → Z ← Y): Z is a common effect. Conditioning on Z creates spurious correlation between X and Y.

Observation vs Intervention

The critical distinction in causal inference is between passive observation and active intervention:

• P(Y | X): Probability of Y given we observe X. This includes all paths (causal and non-causal).
• P(Y | do(X)): Probability of Y if we set X by intervention. This only includes causal paths from X to Y.

When we intervene on X, we "cut" all incoming arrows to X, removing confounding effects and isolating the causal effect.

Confounding and Backdoor Paths

A backdoor path is a non-causal path from X to Y that goes through a common cause. For example, in X ← Z → Y, the path X ← Z → Y is a backdoor path that creates correlation without causation.

To identify the causal effect of X on Y from observational data, we must block all backdoor paths by conditioning on appropriate variables (the backdoor criterion).

Simpson's Paradox

Simpson's paradox occurs when a trend appears in different groups but reverses when the groups are combined. This happens when we fail to account for a confounder:

• Treatment X appears harmful overall: P(Recovery | X) < P(Recovery | ¬X)
• But treatment is beneficial in both severity groups when we condition on severity Z
• The paradox arises because treatment and recovery both depend on severity (a confounder)

Causal graphs make clear when we should condition on Z (to remove confounding) and when we should not (to avoid conditioning on a collider or mediator).

The Do-Calculus

Pearl's do-calculus provides three rules for transforming expressions involving the do-operator, allowing us to determine when causal effects can be identified from observational data. The key insight: interventions break incoming causal arrows, changing the graph structure.

Philosophical Implications

• Causation vs Correlation: Correlation does not imply causation, but causal graphs + interventions (or their observational equivalents) can establish causation.
• Randomized Controlled Trials: RCTs work by randomizing treatment assignment, thereby cutting all arrows into the treatment variable and eliminating confounding.
• Counterfactuals: Causal models enable reasoning about what would have happened under different interventions—the foundation of counterfactual reasoning.
• Free Will and Agency: The do-operator models agency: choosing to make X true is fundamentally different from merely observing X.

Using This Toy

• Select preset DAG structures to explore different causal configurations
• Adjust causal strengths to see how effect magnitudes change
• Compare "No intervention" (observational) with do-interventions to see the difference
• Condition on variables to see when it blocks confounding vs creates spurious associations
• Switch to "Probability Distributions" view to see numerical probabilities