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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