Causal Inference: Causal Structure

What a graph identifies, and what each rule costs when its condition fails

Every value is committed notebook output. The first two examples run on simulations with known effects, which is the only honest way to test whether an identification rule recovers a known answer, and on the Sachs benchmark. The third repeats both methods on national health-survey data, where the reader can grade the result without deferring to anyone.

A is the group’s substance. Four rules, each recovering a known effect when its condition holds — that is the left-hand column, and it is a genuine achievement, because these are questions no estimator can answer on its own. The back-door criterion does not merely resemble the unconfoundedness assumption behind matching; it is that assumption, written as a property of a picture instead of a sentence about treatment assignment.

The right-hand column is the part usually left implicit. Two of the four rules rest on conditions about unobserved variables, so neither can be checked against the data it is applied to, and both were tested here by violating them deliberately. The front door degrades to 134% above the truth, sliding back toward the confounded answer it was brought in to escape. Mediation degrades further: the estimated direct effect goes from 0.50 to 0.06. In both cases the estimator returns a number in every row, with a standard error, in the same table.

B is the result worth carrying out of the group. Adding one control to a correctly specified regression moves a true effect of 2 down to 0.50. The algebra gives exactly one half for this design, so it is structural rather than an unlucky draw, and nothing in the output looks wrong — the coefficient is precise and the model fits better. The instinct that more controls are safer is not merely imperfect here, it is inverted, and this is the one bias in the whole arc that gets worse the more careful an analyst is by conventional standards.

It also generalises further than a lesson about a third variable. Selection into a sample is conditioning on a collider. Surviving long enough to be surveyed, being admitted to hospital, staying in a panel, appearing in an administrative file — each is a common effect of the things being studied, and each induces the same spurious association silently, in data that looks complete and well-measured.

C is the honest limit, and it is why the second page exists. If identification is read off a graph, everything depends on where the graph came from — so the natural question is whether it can be learned. Partly. On the field’s own benchmark, PC achieves perfect precision, which sounds like a triumph until you notice it proposed 7 adjacencies out of 55 possible pairs. It is being reluctant rather than accurate, and what little it commits to is safe.

The sweep is what settles the interpretation. Loosening the test twentyfold adds three false edges and one true one, so the ten missing edges are not near misses waiting for a friendlier threshold — they are absent from the observational distribution, and no tuning recovers them. Two independent implementations in two languages, using two different algorithm families, return the same seven edges. The boundary is in the data.

Which gives the group its ending, supplied by molecular biology rather than econometrics. Sachs et al. reconstructed the full network by intervening — perturbing each protein in turn — not by analysing the observational data harder. The canonical demonstration that causal structure can be learned from data was completed by running experiments, which is this arc’s through-line arriving from an unexpected direction.