# Marginal Distribution of Matrix Normal with Two Inverse Wisharts

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?

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