Generalized linear model - confused by definition [duplicate]

Question

I'm beginning with the regression analysis and I'm quite confused with the generalized linear regression.

I understand, that the ordinary linear models can be described with a formula

$$ y_i = \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_n x_n + \epsilon_i $$

and when using generalized linear regression I'd expect it to be something like this

$$ g\left(y_i\right) = \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_n x_n + \epsilon_i, $$ $g$ being the link function.

But, instead I came across this formula

$$ g\left(E[\mathbf{y}]\right) = \mathbf{x'}\mathbf{\beta} $$

and I don't get, what's the expected value in the left doing there.

Could you, please, explain it a little?

Answer 1

You are correct that regression model is

$$ y_i = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_n x_n + \epsilon_i $$

that can be written differently as

$$ \mu_i = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_n x_n \\ y_i = \mu_i + \epsilon_i $$

or saying it differently

$$ y_i \sim \mathcal{N}(\mu_i, \sigma^2) $$

i.e. we predict conditional mean of $Y$ given $X$ and assume residuals to be normally distributed around it.

and when using generalized linear regression I'd expect it to be something like this

$$ g\left(y_i\right) = \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_n x_n + \epsilon_i, $$

$g$ being the link function.

But, instead I came across this formula

$$ g\left(E[\mathbf{y}]\right) = \mathbf{x'}\mathbf{\beta} $$

and I don't get, what's the expected value in the left doing there.

The above definition is correct. GLM's are generalizations of regression to distributions other then normal and link functions other then identity link. GLM is defined in terms of linear predictor

$$ \eta_i = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_n x_n $$

that is passed through inverse of the link function $g$ to obtain mean

$$ E(Y|X) = \mu_i = g^{-1}(\eta_i) $$

and $\mu_i$ is the conditional mean of $Y$ given $X$ of some probability distribution of choice (e.g. normal, Poisson, Bernoulli etc.).