Let $X \sim N_q(\mu, \Sigma)$ with a Normal-Inverse-Wishart prior on $(\mu, \Sigma)$ i.e. $(\mu, \Sigma) \sim N_q(\mu | m, \Sigma / k) IW_q(\Sigma | \nu, \Lambda)$ where we are using the Inverse Wishart paramaterization that gives $E [\Sigma] = \frac{\Lambda^{-1}}{\nu - q -1}$.
I need the marginal distribution of $X$; I think it should be multivariate $t$ but I'm not sure. I know that if I observed $X_1, ..., X_n$ then the posterior of $\mu$ is multivariate $t$ and that the posterior predictive distribution of $X_{n+1}$ is also multivariate $t$. But I'm not so good with the calculations to get the marginal of a single $X$ under the Normal-Inverse-Wishart and I can't find it derived anywhere.
Hopefully someone knows this off the top of their head; I don't need a derivation although a reference would be good.
