I am looking for the conditional probability of a Wishart distribution, i.e., if I have a Wishart distributed variable $ S \sim W(\Sigma,n), $ where

$$ S = \begin{bmatrix} S_1 \quad S_{12} \\ S_{21} \quad S_2 \end{bmatrix},$$

I would like to know the parameters $\Sigma_\text{cond},n_\text{cond}$ of the Wishart distribution of $S_2$ given $S_1$, i.e.

$$ S_2 \mid S_1 \sim W(\Sigma_\text{cond},n_\text{cond}).$$

This of course assumes that the conditional is also Wishart distributed, which I'm not entirely sure that it's true.

I have looked for this definition in some text books but I could only find the marginals. Can someone help?

  • 1
    $\begingroup$ It will be non-central wishart. I will post details tomorrow! $\endgroup$ – kjetil b halvorsen Mar 10 '17 at 21:00
  • $\begingroup$ I'm wondering why I never thought of this question before. $\endgroup$ – Michael Hardy Mar 11 '17 at 6:42
  • $\begingroup$ The Schur complement $S_2 - S_{21} S_1^{-1} S_{12}$ has a Wishart distribution. Conditioning on $S_1$ is complicated by the question of the conditional distribution of $S_{12}$ (and a fortiori of its transpose $S_{21}$) given $S_1$. But I suspect there's a slick way to do this. $\qquad$ $\endgroup$ – Michael Hardy Mar 11 '17 at 6:51
  • $\begingroup$ I'm vaguely recalling (on this I'm rusty) that the Schur complement $S_2 - S_{21} S_1^{-1} S_{12}$ is independent of $S_1$, but tell me if I'm wrong about that. If so, then the conditional distribution of the Schur complement given $S_1$ does not depend on $S_1$, so it's just the Wishart distribution of the Schur complement. But if this is right, it still doesn't finish off the problem. $\endgroup$ – Michael Hardy Mar 11 '17 at 20:06
  • $\begingroup$ @kjetilbhalvorsen can we see those details? :) $\endgroup$ – Jorge Jesus Mar 15 '17 at 11:23

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