So here I am studying inference. I would like that someone could enumerate the advantages of the exponential family. By exponential family, I mean the distributions which are given as \begin{align*} f(x|\theta) = h(x)\exp\left\{\eta(\theta)T(x) - B(\theta)\right\} \end{align*}
whose support doesn't depend on the parameter $\theta$. Here are some advantages I found out:
(a) It incorporates a wide variety of distributions.
(b) It offers a natural sufficient statistics $T(x)$ according to the Neyman-Fisher theorem.
(c) It makes possible to provide a nice formula for the moment generating function of $T(x)$.
(d) It makes it easy to decouple the relationship between the response and predictor from the conditional distribution of the response (via link functions).
Can anyone provide any other advantage?