# Are all distributions with conjugate priors exponential families?

The Wikipedia page for conjugate priors lists several examples. Of the ones I'm immediately familiar with, all are exponential families. This leads me to wonder whether all families distributions that admit conjugate priors are exponential families.

More explicitly: suppose that $$D$$ is a family of distributions over $$X$$, parameterised by $$\Theta$$, that is, a probability measure $$p(x;\theta)$$ for each $$\theta\in\Theta$$. Suppose $$D$$ admits a conjugate prior. Does it follow that the family $$D$$ is an exponential family? If not, what is a simple counterexample, i.e. a family of distributions that admits a conjugate prior but is not an exponential family?

• For the benefit of readers, since this wasn't completely straightforward to google: "uniform model" refers to a family of distributions on the reals, with a single parameter $a>0$, defined by probability density function $\rho(x;a) = 1/a$ if $0\le x\le a$, and 0 otherwise. It's not an exponential family because the distributions don't all have the same support, and it has a conjugate prior given by the Pareto distribution. Details are given in these notes, for example. Jul 29, 2020 at 9:22