E[Y|D,Z] would be the ultimate goal (predicting the outcome as a joint function of treatment and covariates). This is what is done for instance by standard ML methods as presentend in the beginning, but in this case doesn't provide a good estimator of the treatment effect. I think the approach here is similar to multilinear regression where we first regress D on Z, then we obtain a residual which we regress on Y to isolate the effect of D independently of Z. So the question is rather here : why do we regress D-E[D|Z] on Y-E[Y|Z] instead of Y ? In Multilinear Regression, the first step ensures that the residual will not be correlated to Z, so regressing this residual on Y or Y-E[Y|Z] is equivalent. But here, since the model is semilinear (I think, but perhaps also because we use ML methods), there may be some effect of g(Z) on Y correlated to D even if we take the residual D-E[D|Z]. So we need to evaluate Y-E[Y|Z] to approach the real treatment effect.

@@mathieumaticien Hi the slides have already been removed

@@ruizhenmai1194 why?

Yes, though one can assume that that constant had been partialed out, which would give zero.

no, because D isn't binary, D is continuous. D=m(z) + V, where V is a random variable which does not depend on z. In a perfect trial, D=V, so for example, D might be drawn from a Gaussian distribution with sufficient support to make the inferences you wish to make. You could go further and say that in a "perfect" trial, V is a uniform distribution over some sufficiently large domain. I think here, "perfect" just means "not confounded at all"