= Doubly robust estimation
{wiki}
<Doubly robust estimation> combines two <nuisance function>[nuisance functions] so that either one can be correctly estimated while the other is misspecified. For example, let $A\in\{0,1\}$ indicate treatment, $X$ <covariate>[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 in causal inference>[positivity] and the usual consistency of observed and <potential outcome>[potential outcomes], $\mu_a=\mathbb E[m_a(X)]$ is the <mean> <potential outcome> under treatment $a$.
For candidate <functions> $\widetilde m_a,\widetilde e_a$ with denominators bounded away from zero, the augmented signal is
$$
\phi_a=\widetilde m_a(X)+\frac{\mathbf1_{\{A=a\}}}{\widetilde e_a(X)}[Y-\widetilde m_a(X)].
$$
Conditioning on $X$ shows that
$$
\mathbb E\phi_a-\mu_a=\mathbb E\left[\frac{(\widetilde e_a-e_a)(\widetilde m_a-m_a)}{\widetilde e_a}\right].
$$
The 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 <statistical sample>[sample]; valid <confidence interval>[confidence intervals] require further rate and <moment> assumptions. The <average treatment effect> is $\mu_1-\mu_0$.<Double Debiased Machine Learning for Treatment and Structural Parameters by Victor Chernozhukov and others>[arXiv v6, Section 5.1, equation (5.3), orthogonal average-treatment-effect score]{ref}
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