The Do Operator and Ignorability

The one idea: P(Y | do(X=x)) and P(Y | X=x) look almost identical on the page and mean completely different things — one asks what happens if you reach in and set X, the other just asks what you'd expect to see if you happened to observe X at that value. Observational causal inference only works when you can argue, from outside the data, that the two coincide — and that argument, not any statistical test, is the actual foundation of the whole enterprise.

⚠️ The trap: "I controlled for the confounders I could think of, so my estimate is unconfounded." That feels right and it's wrong. Ignorability — the assumption that treatment is 'as good as random' once you control for the right variables — can never be verified from the data itself. It can only be argued for, using domain knowledge about how treatment actually got assigned, and no amount of statistical machinery substitutes for that argument.

Every lesson so far has quietly leaned on a distinction this lesson finally makes explicit: the difference between seeing and doing. Judea Pearl built a whole notation around it because the difference is easy to state and easy to forget under deadline pressure, and forgetting it is where a shocking share of bad causal claims come from.

Two conditional probabilities that are not the same thing

Ordinary conditional probability, P(Y | X = x), asks: among the cases in my data where X happened to equal x, what's the distribution of Y? It's a pure summary of association. It says nothing about mechanism. If ice cream sales are high, what's the distribution of…