Given a real number $a\ne 0$ and two independent $\mathbb R^d$ valued random variables $X$ and $Y$ with $Y\sim\mathcal N(\mu_Y,\Sigma_Y)$ ($X$'s dsitribution is unknown), I am trying to find a measurable function $f:\mathbb R^d \to \mathbb R^d$ which solves $$\min_{f \text{ measurable}} \mathbb E_{X,Y}\left[|f(X + aY) - Y|^2\right] \tag1$$
Well, if I denote $Z:=X + aY$ it is well known that the solution of $(1)$ is given by $f^* : z \mapsto \mathbb E[Y\mid Z = z]$. Therefore my problem boils down to finding the expression of $f^*$, which I have tried doing in the following way :
Given that $X + aY = z $, we have $\mathbb E[X\mid Z = z] + a\mathbb E[Y\mid Z = z] = z$, i.e.$$f^*(z) = \frac{z-\mathbb E[X\mid Z = z]}{a} $$
But I don't know how to compute $\mathbb E[X\mid Z = z]$. I have seen this and this post on stats.SE which seem to give a solution in case where both $X$ and $Y$ are Gaussian, but in my case $X$ is actually not Gaussian (it can be taken to be a Gaussian mixture if that helps).
Is it possible to find a closed-form for $f^*$ in terms of the moments of $X$ and $Y$ ?