THE BACKDOOR CRITERION

Which variables must you adjust for to read a causal effect off plain observational data? Pearl's back-door criterion is the graphical answer: block every path that sneaks confounding into treatment, touch nothing downstream of treatment, and the adjusted formula equals the interventional truth P(Y|do(X)) — exactly. This instrument builds a confounded structural causal model, adjusts three ways, and lets the truth from do-surgery judge each one.
source Pearl, J. “Causal diagrams for empirical research.” Biometrika 82(4), 669–688 (1995). Rendered, not quoted.
◧ Blue Team · builds & defends
3

The Model

A binary SCM with a hidden confounder U, treatment X, mediator M, outcome Y. Arrows, with known parameters:

u → x   u → y
x → m → y   x → y

U forks into both X and Y: that is the back-door path X ← U → Y. The causal flow is X → Y plus X → M → Y. Only U sits on a path into X.

P(U=1)=.50 · P(X=1|U)=.20/.80 · P(M=1|X)=.30/.70 · P(Y=1|X,M,U)=.10+.30X+.25M+.30U

5

The Lineage

This is the-confounder's cure made precise. The confounder sphere shows the wound — an open back-door leaks bias into every naive comparison. The back-door criterion is the graphical test that says exactly which set Z closes the wound: adjust for these, and the association becomes the effect. It is the workhorse of all observational causal inference.

7

The Witness

Re-runs the identification check live on whatever adjustment set is active: is Z a valid back-door set, and does adjust(Z) equal the do-truth? Trip the tamper in window 6 and this badge flips red.

witness idle
◉ The Machine · back-door adjustment
4

Data In in ↓

Observational samples only — you never see do(X=x), only the joint P(U,X,M,Y) the world hands you, and you cannot observe U unless you name it in Z. Pick the query and the adjustment set:

query 
adjust for 
0

The Panel LIT

criterion: —

Back-door adjustment:  P(Y=1|do X=1) = Σz P(Y=1 | X, z)·P(z)

8

Data Out out ↓

Proven at boot — adjusting for the valid back-door set Z={U} reproduces the interventional truth to machine precision:

booting…
◨ Red Team · attacks & breaks
1

The Adversary

WALL “Just control for everything you measured — more covariates, less bias.”

False. Adjusting for a mediator M blocks part of the real effect. Adjusting for a collider (or a descendant of one) opens a spurious path and manufactures bias where none existed. The criterion is a fence, not a bucket: some variables must be excluded.
2

The Graveyard

"P(Y|X) is the effect of X on Y."

↳ Only when no back-door path is open. Here P(Y=1|X=1)=.815 but the effect gives .725 — confounding by U.

"Adjust for the mediator M to be safe."

↳ M is a descendant of X; the criterion forbids it. It amputates the X→M→Y path — over-control bias.

"Any set that makes X and Y look independent works."

↳ No — the set must block only back-door paths and leave the causal paths intact.

6

The Tamper

The disclosed planted void: swap the adjustment set to the mediator Z={M} — a variable the back-door criterion explicitly forbids (it is a descendant of X). The estimate then blocks part of the true effect and no longer equals the do-truth. The Witness (7) catches it live.