I am working with a Bayesian hierarchical model that has a number of parameters for each experimental unit (6 parameters). I really do not know all that much about them a-priori, but it is quite plausible that they could somehow be correlated.
Thus, I was considering a multivariate normal distribution for the random effects with a pretty uninformative prior on the mean of the random effects and some prior for the covariance matrix. An inverse Wishart with, say, dimension + 1 degrees of freedom (in my example = 7) and the identity matrix as the scale matrix may seem logical, because it is available by default in my software packages and is supposedly relatively uninformative.
I have seen in Gelman's Bayesian Data Analysis book that another option could be to use a separate random effects standard deviation for each parameter in combination with an inverse Wishart with the idea that one could specify more informative priors on the random effects standard deviations (while keeping the correlation structure more uninformative; in addition there's the LKJ prior, which as far as I know is only implemented in Stan.). What I am wondering is in part whether I would really need to do that when I do not know much a-priori and expect to have enough data that hopefully the data would dominate whatever prior I specify (as long as the prior distribution allows sufficient flexibility that this actually happens).