Suppose x is normally distributed with unknown mean and covariance matrix. Is there a set of conjugate priors for the mean and correlation matrix. I know that for covariance matrices one can use inverse Wishart distribution as priors, but what about a correlation matrix?

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    $\begingroup$ Knowing the correlation matrix is equal to $R$ doesn't fully specify the distribution, since there are any number of scalings consistent with that correlation. Is there a particular factorization of the covariance you have in mind, ie, $X \sim$Normal$(\mu, RD)$, for $D$ a diagonal matrix giving variances and $R$ positive definite matrix giving correlations? $\endgroup$ – Andrew M Nov 13 '14 at 8:33
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    $\begingroup$ If you let conjugacy go (because we have [more] efficient sampling algorithms these days), you can use the LKJ distribution, which is a distribution of correlation matrices. $\endgroup$ – Sycorax Nov 14 '14 at 0:29

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