# Linear Model (on X or in $\beta$?)

I'm well aware that when we use the expression "linear model" we are actually making reference to models that are linear on the parameters $$\beta$$. And because of that any polynomial regression will be consider as linear.

Anyway I've been reading "Elements of Statistical Learning" (Hastie, et.al.) and I found a phrase that confused me a little.

Linear regression, linear discriminant analysis, logistic regression and separating hyperplanes all rely on a linear model. It is extremely unlikely that the true function f(X) is actually linear in X. In regression problems, f(X) = E(Y |X) will typically be nonlinear and nonadditive in X, and representing f(X) by a linear model is usually a convenient, and sometimes a necessary, approximation. (p.139)

He states that generally we use, for the sake of simplicity, linear models, but that it's extremely unlikely to have a true function linear in X. And because of that we could use "non linear models" (but non linear in X, like for example polynomial regression).

In my opinion it's a little bit confusing, given that generally we use the terminology "non linear models" to talk about "non linear in parameters".

Am I missing something? Is it usual to use such terminology when referring to polynomial regression?

TL;DR The wording may be confusing, but he means that $$f(X)$$ is a linear function of $$X$$.
By linear model we mean approximating the relation between $$Y$$ and $$X$$ using a function of $$X$$, i.e. $$Y \approx f(X)$$, where the function $$f$$ is a linear function. Linear function is defined as $$f(x) = a + bx$$, where $$a,b$$ are parameters. If the model takes form of a non-linear function, then we call it a non-linear model. Notice that he discusses any machine learning model, not only regression models, and such models can be non-parametric, so "linearity in parameters" does not apply to them at all.