Causal Inference and the Do-Calculus

Takeaway

The do-calculus uses graphical rules to identify causal effects from observational data when interventions are not available.

The problem (before → after)

Before: Correlation confounds causation; conditioning can introduce bias (e.g., colliders).
After: Use DAGs and three do-calculus rules to derive expressions for interventional quantities P(Y | do(X=x)) in terms of observables when possible.

Mental model first

Think of a plumbing system: pipes (edges) carry influence; clamps (conditioning) can stop or open flows. The do-operator replaces sources with fixed valves, letting you trace flow without backwash from confounders.

Just-in-time concepts

DAGs, backdoor/frontdoor criteria, d-separation.
Do-operator: do(X=x) cuts incoming edges into X.
Identifiability: Existence of an expression for P(Y | do(X=x)) using observed distributions only.

First-pass solution

Check backdoor paths and adjust on a valid set Z; if none, try frontdoor with a mediator M that blocks unobserved confounding; otherwise apply do-calculus rules to transform expressions.

Transportability: Move effects across domains with selection diagrams.
Mediation and path-specific effects.
Causal discovery: Learn graph structure under assumptions; then identify effects.

Principles, not prescriptions

Draw the graph first; algebra follows structure.
Adjust only on valid sets; conditioning on colliders biases estimates.

Common pitfalls

Adjusting for descendants of treatment or colliders.
Confusing conditional independence with causal independence.

Connections and contrasts

See also: [/blog/causal-trees], [/blog/double-ml], [/blog/simpsons-paradox].

Quick checks

What is backdoor adjustment? — Conditioning on a set that blocks all backdoor paths from X to Y.
When use frontdoor? — When a mediator M is observed and blocks unobserved confounding.
What is identifiability? — Expressing causal effects purely from observables.