The title might not be perfect. But here goes: Suppose there are 3 variables $(A,X,Y)$. And they have the following dependencies :

$\Pr(Y,A,X)=\Pr(Y\mid A,X)\Pr(X\mid A)\Pr(A)$

$A \rightarrow (X,Y)$

$X \rightarrow Y$

If we only need to estimate $E(Y\mid A)$ I can do it by estimating some marginal mean $\widehat{E}(Y\mid A)$ call this estimator 1. It does not use any information on $X$.

Is there any criterion on $X$ where it would be beneficial to estimate $E(Y \mid A)$ by the law of total expectation?

That would mean to estimate $\widehat{E}(Y\mid A,X)$ and then $\widehat{P}(X\mid A)$ and integrating over the support of $X$:

$$\int_{x\in X}\widehat{E}(Y\mid A,x)\widehat{P}(x\mid A)dx.$$

Call this estimator 2

The question is basically when is preferable to go through the hassle of using $X$? This arises from a predictive model I am trying to estimate. When I will use this model, the only input I can use is $A$ to determine $Y$. But clear relationships exist between $X$ and $Y$, and $A$ and $X$, under what circumstance estimator 2 would be superior to 1?

I tried to think of this through bias and variance of the 2 estimators, but aside from cross validating both techniques, I could reach an analytic solution. Any references to similar stuff is welcome.


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