A quick perplexity popped up in my mind while reading about non-parametric linear regression.
In linear regression, we model our response $\textbf{y} \sim \mathcal{N}(X\beta, \sigma^2I)$ so basically we try to estimate a linear function of the form
$$f_\beta(\textbf{x}_i) =\textbf{x}_{i,1}\beta_1, \dots, \textbf{x}_{i,p}\beta _p$$
while in non-parametric regression we allow more possibilities for the structure of $f$ and the response is modeled as
$$\textbf{y} \sim \mathcal{N}(f(x), \sigma^2I)$$
with $f$ respecting some smoothness assumptions.
What it's not too clear to me is what is the main difference between kernel linear regression and non-parametric one. It is well known that the word linear in linear regression refers to the parameters, so one typically applies a non-linear feature transformation $\phi : \mathbb{R}^p \rightarrow \mathbb{R}^d$ to the features and then searches for some hyperplane fitting the data (brought in higher dimension by the map $\phi$).