Suppose that $X\sim IW_{p}(n,I_p)$ has an inverse Wishart distribution, which probability density function is $$f(X\mid n)\propto |X|^{-\frac{n+p+1}{2}}exp\Big(-\frac{1}{2}tr( X^{-1})\Big),~~\qquad (1)$$ Partition the matrices $X$ with $$X=\pmatrix{X_{11}&X_{12}\\X_{21}&X_{22}},$$ where $X_{ij}$ is $p_i\times p_j$ matrices. From the definition of IW, we know $X_{11}\sim IW_{p_1}(n-p_2,I_{p_1}).$

Qustion: If random $p\times p$ positive matrice $Y$ satisfies $$f(Y\mid n)\propto |Y|^{-\frac{n+p+1}{2}}exp\Big(-\frac{1}{2}tr( Y^{-1})\Big)\Big[\prod_{i=1}^p(\lambda_i-\lambda_j)\Big]^{-1},~~\qquad (2)$$ where $\lambda_1>\lambda_2>\cdots>\lambda_p>0$ are the ordered eigenvalues.

Partition the matrices $Y$ with $$Y=\pmatrix{Y_{11}&Y_{12}\\Y_{21}&Y_{22}}.$$ What is the marginal distribution of $Y_{11}???$


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