# Generalized linear model Gaussian distribution Linear Model

Is a generalized linear model with a Gaussian distribution the same as a linear model?

• If using an identity link, yes, I think it is but not confident enough to post as an answer! Apr 2, 2015 at 13:02
• @tristan: Not necessarily, think e.g. about a linear model that optimizes L1 loss instead of squared loss. The Gaussian GLM with identity link is essentially identical to the corresponding least squares model. Apr 2, 2015 at 13:22
• @MichaelM good point, that's why I didn't add as an answer because I wasn't confident of caveats etc Apr 2, 2015 at 13:23

• What about having unknown variance $\sigma^2$ in the "linear model"? Jul 14, 2021 at 1:27
In the generalized linear model (specially meant for classification problem), the source of non-linearity is activation funtion with linear argument of the form $$\omega^\top x +\omega_0$$. Here, we just need to estimate the parameters $$\omega, \omega_0$$ only and the activation might be user defined.
While, in regression problem, the using least square estimation, the prediction function happens to be a conditional expectation of the form $$E(y|x)$$. If the joint distribution of x and y is Gaussian, then the prediction function $$E(y|x)=\omega_0 + \omega^\top x$$ linear both in the parameters $$\omega, \ \omega_0$$ and in the predictor x.