generalized linear model

In linear regression, we say that the mean of the dependent variable depends on the independent variable in the following way:

\(\mathbb{E}[Y \mid X] = \mu = X\beta\)

The following statistical model is used:

\(Y = X\beta + \epsilon\)

where \(\epsilon \sim \mathcal{N}(0,\sigma^2)\) is Gaussian noise.

For generalized linear models (GLMs), we generalize these two things:

1. \(\mu = g^{-1}(X\beta)\), where \(g\) is called a link function. So now, \(\mu\) is no longer restricted to simply being a linear function of \(X\)
2. the statistical model can vary. That is, once \(\mu\) is set, the dependent variable is not just restricted to a Gaussian distribution, it can be anything from the exponential family