# When to use all 3 variables in a graph to estimate a conditional expectation of 2

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.