Why isn't every nonparametric model with random model design an additive noise model? Let $Y$ be a real random variable and $X$ be a real random vector. In a nonparametric model with additive noise, we assume the relationship
$$Y = f(X) + \epsilon$$
for some unknown regression function $f$ and noise $\epsilon$. This assumption is in contrast to the general nonparametric model, where no assumption about the additivity about the noise is made. Now, I'm wondering why not every model can be written in the former form.
We can always take $f(x) := E[Y \,|\, X = x]$ and $\epsilon = Y - f(X)$. This gives the form
$$Y = f(X) + \epsilon$$
Moreover, we find $E[\epsilon] = 0$ and $E[X\epsilon] = 0$. Am I missing some assumption for the nonparametric regression model with additive noise that is not satisfied here? Otherwise, it seems to me, that the general nonparametric regression model and the nonparametric regression model with additive noise are equivalent.
 A: Edit: Based on some discussion with rkvymvqt from a purely agnostic perspective should always be able to write,
$$Y=f(X)+\epsilon$$
By simply defining $\epsilon = f(X)$ and indeed by using the Doob-Dynkin lemma we can consider $f(x)=E[Y|X=x]$. In some sense, this is just like writing $Y=X+(Y-X)$. I think the issue that we as statisticians are interested in is more in the interpretation and recovery of $f$ from $(Y,X)$. In that case, writing $f$ like this does restrict our interpretation of it and so perhaps it is not as general as finding the "true" $f$ that fits $Y=f(X)$ (without error). Thus, the original answer below is more a comment on how we define the joint relationship for the interpretation of our model. Note, this does not mean that the two models are equivalent because we are recovering two different functions but does mean we can always represent a general model as a nonparametric regression in the case that we do not care about the interpretation of $f$.
Original Answer:
Yes, you are making a critical assumption that the error is additive. That is a functional form assumption. You are defining the conditional expectation to be,
$$E[Y|X=x]=f(x)$$
This is called a nonparametric regression model or equivalently an additive noise model. We could just as easily believe that our model should have a multiplicative error structure,
$$Y=f(X)\epsilon$$
See for example this paper. In this case we would have,
$$y_i = E[y_i|X=x_i]\epsilon_i$$
In the most general sense the nonparametric model is written as,
$$\{P_f:f\in\mathcal{F}\}$$
Where $\mathcal{F}$ is some infinite-dimensional parameter space and our data was generated by the probability distribution $P_f$ for some parameter $f$.
