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Let's say, I want to establish a relation between the distribution of $Y$ and $p$ covariates. I denote the covariates with $x_1,x_2,...,x_p$. Then, I denote the linear predictor with $\eta$ , where $\eta = \beta_0 + \beta_1 x_1 + ~... ~+\beta_p x_p$ .

Thus, $E \left[Y|X \right]=\mu$, which is connected with the linear predictor by the response function $h$, where $\mu=h(\eta)$. Finally, I call $g$ the inverse of $h$, where $g= h^{-1}$. It must be noted that $\eta=g(\mu)$, and $g$ is called the link function.

Why does the link function $g$ make the model linear? The question seems a bit trivial, but is the answer that $\eta=g(\mu)$ is just a definition, in the sense that the linear predictor is linear because $g$ is a linear function?

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  • $\begingroup$ It's called linear because the cheeky statisticians apply the link function on the left side of the equation, so viewed from that angle, the right side (linear predictor) remains a polynomial. However, when considering the effect of parameter changes on the expectation value of the response, a GLM is not linear. $\endgroup$ Commented Nov 27, 2022 at 19:14

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Some clarifications:

$\bullet$ The linear predictor

$$\eta_i =\sum_{j=1}^p\beta_jx_{ij}\tag 1\label 1$$

is linear in the parameters.

$\bullet$ $g(\cdot), $ aka the link function that links $\eta_i$ to $\mu_i$ by $\eta_i=g(\mu_i), $ i.e., $\mu_i\mapsto \mathbf x_i^\top\boldsymbol\beta$, is monotonic and differentiable.

So, $g\left(\mathbb E[Y]\right) $ is linear in the parameters. This is the basis of GLM.

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