# Generalized Linear Models - What's special about the exponential family?

In Generalized Linear Models the conditional distribution of the response variable has to belong to the exponential family. Why is this restriction important? What property would a regression model lose if we chose a distribution outside the exponential family?

• sufficiency of the maximum likelihood estimate is one one thing you will lose. Nov 19 '16 at 14:30
• Theory and algorithms are easier ti develop inside the exponential family. But you can go outside it, and it is done nowadays---see the R VGAM package, doing just this. Nov 19 '16 at 14:36

Jaynes makes the argument that when you leave the exponential family, your estimators cease to be sufficient statistics. If a statistic is sufficient for a parameter then $\Pr(t|\theta)=\Pr(X|\theta)$. Implying that the information in $t$ is the same as in the sample $X$. Bayesian methods always use all the information in $X$. Non-Bayesian methods use a statistic. If that statistic contains the same information then the estimator will be no worse.