Targeted Maximum Likelihood for ATE, why not just GLM for ATE? I have been attempting to learn targeted maximum likelihood (TMLE), which is a struggle. The estimand of interest is often the average treatment effect. If we have an outcome $Y$, intervention $A$ and a set of covariates $W$ this is defined as,
$$ATE = $$
$$E_W( E(Y|A = 1, W) - E(Y | A = 0, W)) = $$
$$ E_W( E(Y|A = 1 ,W)) - E_W( E(Y|A = 0,W))$$
where $E_W$ denotes the expectation with respect to the distribution of the covariates. Why do we need targeted maximum likelihood here? For any GLM I can estimate $E_W( E(Y|A = 1,W))$ by first just finding $E(Y|A = 1,W)$ explicitly from the GLM (similarly for $E(Y|A = 0,W)$)  and marginalizing out over the distribution of my covariates. What is TMLE getting me here that traditional MLE of a GLM is not?
 A: The answer is double robustness. Like all doubly-robust methods, with TMLE you get two chances to get the model correct and the treatment effect estimate is consistent if either is correct (or approach at a given rate). With g-computation using a GLM, you only get one chance to get the model right. If that model is wrong (and it almost certainly is), then your treatment effect estimate is not consistent. You can use flexible machine learning methods with TMLE that are more likely to approximate the true form of either the treatment and outcome model. Indeed, given the inability of an outcome model GLM to even plausibly capture the true outcome model, one should almost never use it to estimate treatment effect, whereas TMLE should be among your first choices.
A: If the dimension of $W$ is large, computing the MLE of the whole GLM and then marginalising need not give the efficient estimator of the ATE.  From an asymptotic viewpoint, suppose the dimension of $W$ grows fast enough that the whole parameter vector of the GLM is not estimable at $\sqrt{n}$ rate. It's still possible that the ATE is estimable at $\sqrt{n}$ rate, and that the plug-in estimator you describe isn't efficient.
Heuristically, efficient estimation of the GLM (by maximum likelihood or any other way) involves getting the bias:variance tradeoff right for the whole parameter vector. But for causal inference what you care about is a particular one-dimensional functional of the parameter vector. It turns out that the optimal tradeoffs can be different. Targeted MLE focuses on estimating the one-dimensional parameter of interest, and accounts for the need to estimate the other parameters by considering the one-dimensional submodel of the GLM in which estimation of the ATE is hardest.
If the dimension of $W$ is small enough that the whole GLM can be estimated well, the simple plug-in estimator will be efficient.  This is what you get with an asymptotic argumen that treats the dimension of $W$ as fixed as the sample size increases.
See also this answer
