I'm studying prior choice, and as far as I understand, when more than one parameter in a distribution is unknown, it is both possible to place one prior for each of the parameters in the likelihood, as well as to place a joint prior over all of said parameters.
For example, for a normal distribution $N(\mu,\sigma^2)$, it is possible to place, say, a normal prior on $\mu$ and another normal prior on $\sigma^2$, but it is also possible to place a bivariate normal distribution as a prior on both $\mu$ and $\sigma^2$.
One of the reasons for using joint priors, I'm guessing, is that we can construct conjugate joint priors for some distributions. However, when not using a conjugate prior, like above, what is the difference between using a joint prior vs separate priors? Is it to encode our belief in the correlation between the parameters? Beyond that, is there any reason to prefer joint priors over separate priors?