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?
Best Answer
I think your list of advantages effectively answers your own question, but let me offer some meta-mathematical commentary that might elucidate this topic. Generally speaking, mathematicians like to generalise concepts and results up to the maximal point that they can, to the limits of their usefulness. That is, when mathematicians develop a concept, and find that one or more useful theorems apply to that concept, they will generally seek to generalise the concept and results more and more, until they get to the point where further generalisation would render the results inapplicable or no longer useful. As can be seen from your list, the exponential family has a number of useful theorems attached to it, and it encompasses a wide class of distributions. This is sufficient to make it a worthy object of study, and a useful mathematical class in practice.
This class has various good properties in Bayesian analysis. In particular, the exponential family distributions always have conjugate priors, and the resulting posterior predictive distribution has a simple form. This makes is an extremely useful class of distributions in Bayesian statistics. Indeed, it allows you to undertake Bayesian analysis using conjugate priors at an extremely high level of generality, encompassing all the distributional families in the exponential family.