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Say I have a Matrix-Normal distribution and two Inverse Wishart Distributions

$$X \sim MN_{p\times n}(0, \Sigma, \Omega)$$ $$ \Sigma \sim IW(a, A) $$ $$ \Omega \sim IW(b, B)$$ where $a$ and $b$ are degrees of freedom and $A$ and $B$ are scale matrices.

I know that the marginal distribution of $p(X, \Omega)$ is matrix-T and given by $ T_{p\times n} (a, 0, A, \Omega)$. What is the marginal distribution of $p(X)$?

In other words, what is the marginal distribution when you marginalize over both inverse Wishart random variables? Is there a name for this distribution?

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  • $\begingroup$ Did this end up having a nice posterior distribution? That is, is this conjugate? $\endgroup$
    – shabbychef
    Commented Feb 13, 2020 at 6:34
  • $\begingroup$ I think this is not going to have a nice form because the Kronecker product of Inverse Wisharts is not a known distribution. $\endgroup$
    – shabbychef
    Commented Feb 13, 2020 at 6:35
  • $\begingroup$ Agreed. I doubt its nice, but this remains an outstanding question. I don't have an answer yet. $\endgroup$
    – jds
    Commented Feb 17, 2020 at 13:56

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