Suppose we have data that is Normally distributed $\mathcal{N}(\mu,\,\sigma^{2})$ with known mean and unknown variance . Then by conjugate priors, a Gamma can be placed on $\sigma^2$ to infer it.
I came across this paper ($\textit{Prior distributions for variance parameters in hierarchical models}$ - $\textit{Andrew Gelman}$) that advocates uniform prior though it is for hierarchical models. Which prior should I use? Also how do I calculate posterior with likelihood of N Normally distributed data samples and a uniform prior on variance?