A family of Probability distributions, commonly used in Generalized linear models

See video.

A probability distribution is in the exponential family if it has the form

$p(y;\eta) = b(y) \exp{(\eta^T T(y) -a(\eta))}$

where $\eta$ is the natural parameter, and $T(y)$ is the sufficient statistic. Most often $T(y)=y$.

The Bernoulli and the Gaussian distributions are both examples.

Exponential dispersion family

$p(y;\theta,\psi) = h(y;\psi)\exp{(\frac{b(\theta)^TT(y)-A(\theta)}{d(\psi)})}$

There is canonical parametrization, and mean parametrization.

Estimating $\psi$ from data. Can do that even when we expect $\psi=1$, and we get a goodness-of-fit test.