Inverse Wishart

Assume that $$X\sim \operatorname{IW}_p(n,\Sigma)$$ has an inverse Wishart distribution, which probability density function is $$f(X\mid n,\Sigma)=C(n,\Sigma)|X|^{-\frac{n+p+1}{2}} \exp\Big(-\frac{1}{2} \operatorname{tr}(\Sigma X^{-1})\Big), \tag 1$$ where $$C(n,\Sigma)$$ is a constant, only depends on $$n$$ and $$\Sigma.$$

Partition the matrices $$X$$ and $$\Sigma$$ conformably with each other $$X=\pmatrix{X_{11}&X_{12}\\X_{21}&X_{22}},~~\Sigma=\pmatrix{\Sigma_{11}&\Sigma_{12}\\\Sigma_{21}&\Sigma_{22}}.$$ where $$X_{ij}$$ and $$\Sigma_{ij}$$ are $$p_i\times p_j$$ matrices.

How to prove $$X_{11}\sim \operatorname{IW}(n-p_2,\Sigma_{11})$$, by the probability density function of $$X$$ in (1) directly, not the definition of $$\operatorname{IW}$$ or Wishart distribution???

• this would be rather surprising if you proved something about $IW$ without even using its definition. Jan 29 '19 at 7:56
• @DimaPasechnik : I'm guessing the poster meant how do you prove it by using the density function on line $(1),$ without using anything about sums of products of certain random matrices. Feb 7 '19 at 20:48
• Might you have meant $X_{11} \sim \operatorname{IW}_{p_1}(n, \Sigma_{11}) \text{ ?} \qquad$ Feb 7 '19 at 20:50
• A pdf is always with respect to some measure. Which measure in this case? How can one integrate? Is it just $$\int f(x) \prod_{i,\,j} dx_{i,j}$$ over the whole space of positive-definite matrices $x$, where $x_{i,j}$ is the $i,j$ entry in $x$? That would mean Lebesgue measure $dx_{i,j}$ on each entry, and the bounds of integration would be --- complicated. This needs some explanation. $\qquad$ Feb 7 '19 at 21:43
• @Michael Hardy Yes. $\int f(x) \prod_{i,j} dx_{i,j}$. I want to know how to prove it by using the density function on the line (1), without using anything about sums of products of certain random matrices. Feb 8 '19 at 23:22