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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???

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  • $\begingroup$ this would be rather surprising if you proved something about $IW$ without even using its definition. $\endgroup$
    – Dima Pasechnik
    Jan 29, 2019 at 7:56
  • $\begingroup$ @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. $\endgroup$
    – Michael Hardy
    Feb 7, 2019 at 20:48
  • $\begingroup$ Might you have meant $X_{11} \sim \operatorname{IW}_{p_1}(n, \Sigma_{11}) \text{ ?} \qquad$ $\endgroup$
    – Michael Hardy
    Feb 7, 2019 at 20:50
  • $\begingroup$ 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. $\qquad$ $\endgroup$
    – Michael Hardy
    Feb 7, 2019 at 21:43
  • $\begingroup$ @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. $\endgroup$
    – Xiaopai Song
    Feb 8, 2019 at 23:22

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