What does an augmented IPW (doubly robust) estimator add over plain IPW?

## Two routes to the same estimand There are two obvious ways to adjust for measured confounders. You can model **treatment** — estimate `e(X) = P(T = 1 | X)` and reweight — or you can model the **outcome** — estimate `m1(X) = E[Y | T = 1, X]` and `m0(X) = E[Y | T = 0, X]`, predict both potential outcomes for everyone, and average. Each is consistent only if its own model is right. Choosing between them is a bet on which model you trust. Augmented inverse probability weighting refuses the bet by using both. ## The estimator For the mean outcome under treatment, AIPW averages over all n units the quantity `m1(X_i) + T_i * (Y_i - m1(X_i)) / e(X_i)` and symmetrically for the control mean, using `m0` and `1 - e(X)`. The effect estimate is the difference of the two means. Read the formula as **prediction plus weighted correction**. The first term is the outcome model's guess for every unit, treated or not. The second term takes the treated units' residuals — how wrong the outcome model was on the people you actually observe under treatment — and spreads them back over the population using inverse probability weights. ## Why it is doubly robust Suppose the **outcome model is correct**. Then the residuals `Y - m1(X)` have conditional mean zero among treated units at every `X`, so the correction term averages to zero no matter what weights you attach. The estimator collapses to the outcome-model answer and is consistent even if `e(X)` is nonsense. Now suppose the **propensity model is correct** but the outcome model is not. Rewrite the same expression as the ordinary IPW term `T*Y/e(X)` minus a term involving `(T - e(X)) * m1(X) / e(X)`. Because `T - e(X)` has conditional mean zero when the propensity model is right, that second term averages to zero, leaving the consistent IPW estimator. The wrong outcome model does not bias the answer; it only changes the variance. So the bias of AIPW is governed by a *product* of the two models' errors. Either factor being zero kills the bias. That is the whole idea, and it is the sentence to say in an interview: **you get two chances to be right and only need one.** ## Efficiency, not just robustness When both models are correct, AIPW attains the semiparametric efficiency bound — no other estimator relying on the same assumptions has smaller asymptotic variance. Even when only the propensity model is right, the outcome model usually acts as a variance-reducing control, so AIPW is typically tighter than plain IPW. ## Modern practice: machine-learned nuisances and cross-fitting Because the bias depends on the *product* of the two errors, each model may converge more slowly than the usual root-n rate as long as the product converges faster. That is what licenses flexible machine-learned models for `e(X)` and `m(X)` while still getting valid confidence intervals for the effect. Two conditions come with it. First, **cross-fitting**: fit the nuisance models on one split of the data and evaluate the estimating equation on the held-out split, rotating through folds, so that overfitting in a flexible model does not leak into the effect estimate. Second, the nuisance models must be regularised sensibly — a flexible treatment model that separates the arms manufactures near-zero scores and destroys stability. ## Where it disappoints 1. **Extreme weights still hurt.** The correction term divides by `e(X)`. If some scores are near zero the augmentation term can be wildly variable, and AIPW can have higher variance than a trimmed plain IPW estimate. 2. **Both models wrong is not covered.** Double robustness protects against one failure, not two. With both nuisances mildly misspecified and poor overlap, the doubly robust estimate can be worse than either single-model estimate. 3. **It does not buy an assumption.** No amount of augmentation addresses unmeasured confounding or a covariate region with essentially no treated units. Both routes rest on the same identification story. 4. **It is harder to explain.** For a stakeholder audience, a well-diagnosed weighted estimate with clear balance evidence may communicate better than a two-model estimator that nobody in the room can sanity-check. ## How to answer the question Name the two nuisance models, state the guarantee precisely (consistent if either is correct, efficient if both are), give the prediction-plus-weighted-residual reading of the formula, and finish with the honest limits: overlap problems and simultaneous misspecification are still fatal.