# When sampling distribution and posterior distribution under uniform prior belong to the same family?

This is similar to conjugate prior, except that here I consider if it is possible that a sampling distribution and the posterior distribution under uniform prior belong to the same family of distributions?

In other words, when viewing $p(x|\theta)$ as the likelihood function of $\theta$ given $x$, and then normalize it to be a distribution for $\theta$, are there examples where $p(x|\theta)$ and the distribution for $\theta$ belong to the same family?

More generally, are there examples for priors other than uniform and sampling distributions, where the posteriors fall into the same family of distributions as the sampling distributions?

Thanks!

1. $x$ is normally distributed around mean $\theta$. The likelihood for $\theta$ is a normal distribution with mean $x$.
2. $x$ is Gamma distributed with rate $\theta$. The likelihood for $\theta$ is a Gamma distribution with rate $x$.