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An augmented inverse-probability-weighted estimator averages a regression prediction plus an inverse-probability-weighted residual. In the treatment setting of doubly robust estimation, let ${\hat{m}}_a(X_i)$ estimate the conditional outcome mean and ${\hat{e}}_a(X_i)$ the probability of receiving treatment $a$. ThenThe first term predicts outcomes for everyone; the weighted residual corrects prediction errors using observed outcomes from the corresponding treatment group. The conditional-expectation calculation in doubly robust estimation makes the remaining bias a product of outcome-regression and propensity-score errors. An exactly correct outcome regression makes the residual correction mean zero even with an incorrect propensity score; a correct propensity score makes that correction repair the regression bias.

The nuisance estimates are commonly obtained by cross-fitting, with each observation evaluated using functions trained on other observations. Both treatment probabilities must remain positive on the population being studied, and small fitted probabilities can make the variance very large. Double robustness concerns consistency under one correct nuisance model; it does not automatically deliver valid finite-sample or asymptotic confidence intervals.

Doubly robust estimation combines two nuisance functions so that either one can be correctly estimated while the other is misspecified. For example, let $A\in \{0,1\}$ indicate treatment, $X$ covariates, $m_a(X)=\mathbb{E}[Y\mid A=a,X]$, and $e_a(X)=\mathbb{P}(A=a\mid X)$. Under conditional exchangeability, positivity and the usual consistency of observed and potential outcomes, ${\mu }_a=\mathbb{E}[m_a(X)]$ is the mean potential outcome under treatment $a$.

For candidate functions ${\tilde{m}}_a,{\tilde{e}}_a$ with denominators bounded away from zero, the augmented signal isConditioning on $X$ shows thatThe bias vanishes if either nuisance function is correct. This algebra explains Double robustness and the second-order product bias. It does not protect against both models being wrong, failed Causal identification, or nearly zero denominators. cross-fitting helps when flexible functions are estimated from the same sample; valid confidence intervals require further rate and moment assumptions. The average treatment effect is ${\mu }_1-{\mu }_0$.^[ref]

A nuisance function is an unknown function needed to identify or estimate a target statistical parameter but not itself the primary estimand. In treatment-effect estimation, examples are the propensity score $e(x)=\mathbb{P}(A=1\mid X=x)$ and the outcome-regression functions $m_a(x)=\mathbb{E}[Y\mid A=a,X=x]$.

These functions can be infinite-dimensional even when the target is one number. Estimating them poorly can bias that target, so nuisance does not mean irrelevant. Doubly robust estimation and Neyman orthogonality control certain nuisance errors, while cross-fitting separates training observations from score evaluation. None of these devices removes the identification assumptions needed to interpret an observational treatment comparison causally.^[ref]