Seminar Notes — August 17, 2026

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Topic

Modern Perspectives on Doubly Robust Estimation

Date: August 17, 2026
Format: Seminar discussion
Presenter: Yunhao Zheng

1. Motivation

A recurring problem in causal inference is that the estimand may be identifiable under relatively weak assumptions, but practical estimation requires nuisance quantities such as

[ \mu_a(x)=\mathbb E(Y\mid A=a,X=x) ]

and

[ e(x)=P(A=1\mid X=x). ]

If either nuisance model is misspecified, a conventional plug-in estimator may become inconsistent.

Doubly robust methods address this problem by combining the outcome regression and propensity score models in such a way that consistency can survive misspecification of one of them.

2. Main estimand

Consider the average treatment effect

[ \psi = \mathbb E\left[ Y(1)-Y(0) \right]. ]

Under consistency, exchangeability,

[ Y(a)\perp A\mid X, ]

and positivity,

[ 0<P(A=1\mid X)<1, ]

the estimand is identified by

[ \psi = \mathbb E[ \mu_1(X)-\mu_0(X) ]. ]

3. AIPW estimator

The augmented inverse probability weighted estimator can be written as

[ \hat\psi = \frac1n \sum_{i=1}^n \left[ \hat\mu_1(X_i) - \hat\mu_0(X_i) + \frac{A_i}{\hat e(X_i)} {Y_i-\hat\mu_1(X_i)} - \frac{1-A_i}{1-\hat e(X_i)} {Y_i-\hat\mu_0(X_i)} \right]. ]

The important point is that this is not merely an arbitrary combination of regression adjustment and IPW.

It arises naturally from the efficient influence function.

4. Main intuition

The key seminar discussion was:

Doubly robust estimation is better understood as bias correction than as “using two models.”

The plug-in estimator

[ \frac1n\sum_i { \hat\mu_1(X_i)-\hat\mu_0(X_i) } ]

contains bias caused by errors in estimating the conditional mean.

The augmentation terms

[ \frac{A}{e(X)}{Y-\mu_1(X)} ]

and

[ \frac{1-A}{1-e(X)} {Y-\mu_0(X)} ]

act as first-order corrections.

This perspective connects AIPW directly to influence-function-based estimation.

5. Why “double robustness” occurs

Let

[ \hat\mu_a-\mu_a ]

denote the outcome-model error and

[ \hat e-e ]

the propensity-score error.

After the first-order correction, the remaining bias is approximately a product of nuisance errors:

[ \operatorname{Bias} \approx (\hat\mu-\mu)(\hat e-e). ]

Therefore, if either

[ \hat\mu=\mu ]

or

[ \hat e=e, ]

the leading bias disappears.

This is the algebraic origin of double robustness.

6. Connection to modern DML

The more modern interpretation is stronger than classical double robustness.

Suppose

[ |\hat\mu-\mu| = o_p(n^{-1/4}) ]

and

[ |\hat e-e| = o_p(n^{-1/4}). ]

Then

[ |\hat\mu-\mu| \, |\hat e-e| = o_p(n^{-1/2}), ]

so the nuisance-estimation error becomes negligible relative to the parametric (n^{-1/2}) scale.

This allows flexible machine-learning estimators to be used for nuisance functions while retaining asymptotically normal inference.

Cross-fitting is then used to weaken empirical-process restrictions caused by estimating and evaluating nuisance models on the same observations.

7. Discussion points

7.1 DR is not the same as efficiency

Double robustness concerns consistency under nuisance-model misspecification.

Semiparametric efficiency concerns attaining the smallest asymptotic variance within a statistical model.

These are related but conceptually different properties.

7.2 AIPW and one-step estimation

AIPW can be understood as a one-step estimator:

[ \hat\psi_{\mathrm{one-step}} = \psi(\hat P) + P_n D_{\hat P}, ]

where (D_{\hat P}) is the estimated influence function.

This interpretation is often more useful than memorizing the AIPW formula.

7.3 The real object is the score / influence function

A useful conceptual workflow is

[ \text{Estimand} \rightarrow \text{Identification} \rightarrow \text{EIF} \rightarrow \text{Estimator} \rightarrow \text{Inference}. ]

Rather than beginning with an estimator and asking why it works, derive the relevant influence function first.

8. Questions that remain open

1. When exactly does Neyman orthogonality imply double robustness?
2. Is every doubly robust estimator associated with an orthogonal score?
3. How should we distinguish EIF-based AIPW from more general orthogonal estimating equations?
4. What changes when the target parameter is not pathwise differentiable?
5. How does TMLE differ conceptually from one-step estimation once both use the same EIF?

9. My takeaway

The most useful conceptual shift from today’s seminar is:

AIPW should not primarily be viewed as “outcome regression + IPW.” It should be viewed as a first-order debiasing procedure derived from the geometry of the statistical model.

The classical “two models, one may be wrong” interpretation explains robustness.

The influence-function interpretation explains why the estimator has this form, how to generalize it to new estimands, and why the same structure appears throughout modern semiparametric statistics and double machine learning.

References

• Robins, Rotnitzky, and Zhao. Semiparametric regression for repeated outcomes.
• van der Laan and Robins. Unified Methods for Censored Longitudinal Data and Causality.
• van der Vaart. Asymptotic Statistics.
• Chernozhukov et al. Double/debiased machine learning for treatment and structural parameters.

Last updated: August 17, 2026.