Assume that $X\sim \mathcal{N}(0,\sigma_X^2)$ and $\epsilon \sim \mathcal{N}(0,\sigma_\epsilon^2)$ are independent and $k\in \mathbb{N}$.
Define $Y:= \sum_{i=1}^k \beta_i X^i + \epsilon$, where $\beta_1,...,\beta_k$ are real numbers.
Is it possible to derive $E(X|Y)$ for general $\beta_1,...,\beta_k\in \mathbb{R}$?
Do you know any large function classes $\mathcal{F} \subset \{f:\mathbb{R}\to \mathbb{R}\}$, such that we are able to derive $E(X|f(X)+\epsilon)$ when $f\in \mathcal{F}$?