  Covariance matrix estimation arises in multivariate problems including
multivariate normal sampling models and regression models where random effects
are jointly modeled, e.g. random-intercept, random-slope models. A Bayesian
analysis of these problems requires a prior on the covariance matrix. Here we
assess, through a simulation study and a real data set, the impact this prior
choice has on posterior inference of the covariance matrix.
  Inverse Wishart distribution is the natural choice for a covariance matrix
prior because its conjugacy on normal model and simplicity, is usually
available in Bayesian statistical software. However inverse Wishart
distribution presents some undesirable properties from a modeling point of
view. It can be too restrictive because assume the same amount of prior
information about every variance parameters and, more important, it shows a
prior relationship between the variances and correlations.
  Some alternatives distributions has been proposed. The scaled inverse Wishart
distribution, which give more flexibility on the variance priors conserving the
conjugacy property but does not eliminate the prior relationship between
variances and correlations. Secondly, it is possible to fit separate priors for
individual correlations and standard deviations. This strategy eliminates any
prior relationship within the covariance matrix parameters, but it is not
conjugate and therefore computationally slow.
