# Generalized Linear Models

Let the outcome belong to $[Y_1,Y_2]$. We say that Expert $\theta$'s prediction at step $t$ is denoted $\xi^\theta_t$ and is equal to

$\displaystyle{\xi_t^\theta = Y_1+(Y_2-Y_1)\sigma(\theta'x_t).}$

Here $\sigma: \mathbb{R}\to\mathbb{R}$ is a fixed \emph{activation function}. We have $\sigma:\mathbb{R}\to [0,1]$ in all the cases except linear regression (see below). If the range of the function $\sigma$ is $[0,1]$, the experts necessarily give predictions from $[Y_1,Y_2]$.

• Logistic activation function is $\sigma(z) = \frac{1}{1+e^{-z}}$.
• Probit activation function is
$\displaystyle{\sigma(z) = \Phi(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^z e^{-v^2/2} dv,}$

where $\Phi$ is the cumulative distribution function of the normal distribution with zero mean and unit variance.

• Complementary log-log (comlog) activation function $\sigma(z) = 1-\exp (-\exp(z))$.

Generalized Linear Models are often used for classification purposes.

### Bibliography

• McCullagh, P., Nelder, J. Generalized Linear Models. Chapman & Hall/CRC, second edn. 1989.