The Do-Operator and Identification

\(P(Y \mid X = x)\) and \(P(Y \mid do(X = x))\) are different quantities and the notation exists because they are routinely confused. The first is the distribution of \(Y\) among units that happened to have \(X = x\). The second is the distribution of \(Y\) if you reached in and set \(X = x\) for everyone, leaving the rest of the system to respond.

The classic illustration: among people who take a particular medication, recovery is worse than among people who do not, because sicker people take it. \(P(\text{recover} \mid \text{drug})\) is low. \(P(\text{recover} \mid do(\text{drug}))\) can be high. Both are correct answers to different questions, and only the second is the one a prescriber needs.

What intervention does to the graph

\(do(X = x)\) corresponds to graph surgery: delete every arrow into \(X\), since \(X\) is now set by you rather than by its usual causes, and set \(X = x\). Everything downstream is unchanged, because intervening on \(X\) does not alter how \(Y\) responds to its inputs.

This makes the asymmetry concrete. Conditioning is filtering rows of the observed distribution; intervening is generating a new distribution from a modified model. In a randomised experiment the two coincide for the randomised variable, because randomisation is graph surgery, physically performed. That is the sense in which an experiment is not a clever analysis of observational data but a different data-generating process.

Identification, and the adjustment formula

A causal quantity is identified if it can be written as a function of the observational distribution. When a set \(Z\) satisfies the backdoor criterion, the answer is the adjustment formula:

Read the two pieces separately. Inside the sum, you condition on \(Z\), which is what a regression does. Outside, you weight by the marginal distribution of \(Z\), not its distribution among the treated. That second part is what converts a conditional association into a causal effect: it asks what would happen if the whole population, with its actual covariate mix, were treated.

Identification is logically prior to estimation and independent of sample size. If the effect is not identified, more data estimates the wrong quantity with tighter confidence intervals. This ordering, settle identification, then estimate, is the single most useful discipline the framework imposes.

The front door, and why it matters

When a confounder is unmeasured, the backdoor is unavailable, and the usual conclusion is that nothing can be done. The front-door criterion shows that is too pessimistic. If the effect of \(X\) on \(Y\) runs entirely through a measured mediator \(M\), and no unmeasured confounder affects \(M\) directly, the effect is identified by combining \(P(M \mid do(X))\) and \(P(Y \mid do(M))\) in two stages.

The conditions are demanding and rarely hold cleanly in practice. What matters is the general lesson: identifiability is a property of the graph, not of whether you can name a confounder, and the complete do-calculus can decide it for any graph and any query.

When it breaks

Identification is relative to a graph you asserted. The formula is correct given the structure. If the structure is wrong, the formula computes a well-defined quantity that is not the causal effect, with no warning of any kind.

Positivity is a separate requirement. The adjustment formula needs \(P(X = x \mid Z = z) > 0\) for every \(z\) with positive probability: some units at every covariate value must plausibly receive each treatment. If a subgroup is always treated, no data exist about what happens when it is not, and the estimator is extrapolating rather than adjusting.

High-dimensional \(Z\) makes the sum a modelling problem. With many covariates the empirical version of the adjustment formula has no cell counts to work with, so it becomes a regression or a weighting estimator, and correct specification re-enters through the back door. Double machine learning exists precisely to keep flexible nuisance models from contaminating the causal estimate (Chernozhukov et al., 2018, Econometrics Journal 21(1), C1-C68).

Not every question is a \(do\) question. Counterfactuals about a specific unit that already received treatment, "would this user have converted had they not seen the banner", sit on a higher rung than interventions and generally require assumptions beyond the graph. Interventional identification does not deliver them.