###### tags: `one-offs` `exponential families`

# Convenient Exponential Families

**Overview**: In this note, I consider why exponential families are often treated as being 'nice' and 'convenient', and discuss some potential confounding factors which are involved in this perception.

## Discussion

I occasionally reflect on the phenomenon that exponential families are usually taught as being 'easy to work with'. In some sense, it's very true, since there are a few examples of exponential families which are extremely friendly (Gaussian, Gamma, Beta, (Negative) Binomial, Poisson), with which one can get quite a lot done, and which are frequently relevant in practical scenarios.

However, with the benefit of hindsight, and experience with thinking about 'intractable' statistical problems, this can sometimes seem like an odd impression. I say this because in those friendly examples, only some of the simplicity is directly attributable to the exponential family structure; there are other relevant ingredients in the mix. Indeed, a generic exponential family whose normalising factor *doesn't* come in closed form is almost as likely to give rise to an NP-hard inference problem as anything else.

A related observation, to which I also return periodically, is that for these friendly examples, the other active ingredient is a connection to divisibility properties and Lévy processes. There is almost a form of duality at play here: exponential families are nice for joint and conditional laws, and Lévy processes are nice for marginal laws. It is thus not so surprising that the intersection of the two properties is so fertile for mathematics.