Suppose $\begin{bmatrix} K_{11} K_{12}\\K_{12}^T K_{22} \end{bmatrix}\sim\mathcal{IW}\left(\eta,\begin{bmatrix} \Sigma_{11} \Sigma_{12}\\\Sigma_{12}^T \Sigma_{22} \end{bmatrix}\right)$.

  1. What is the conditional distribution of $K_{11}|K_{22}$ and $K_{12}|K_{22}$
  2. What is the expectation of $K_{12}^TK_{22}^{-1}$

closed as unclear what you're asking by Stefan Kohl, Steven Sam, Alexandre Eremenko, John Pardon, S. Carnahan Jul 22 '14 at 12:59

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  • $\begingroup$ the question "what is..." has an answer starting from the definition of the Inverse Wishart distribution, but I can't see how this would provide a simple closed-form expression, and I presume that's your real question. – $\endgroup$ – Carlo Beenakker Mar 13 '14 at 9:18
  • $\begingroup$ I was certainly looking for closed form solutions. Not sure what you mean by "what is..." $\endgroup$ – sachinruk Mar 14 '14 at 6:05
  • $\begingroup$ well, since you know the probability distribution of the full matrix, the marginal distribution of a set of matrix elements is obtained upon integration, and then the conditional distribution follows as well; so you know "what is the conditional distribution", it's just that you will not be able to carry out these integrations with paper and pencil to arrive at some tractable expression. $\endgroup$ – Carlo Beenakker Mar 14 '14 at 9:03

Have a look at the book Multivariate Statistics : A Vector Space Approach

by Morris L. Eaton, Chapter 8 pages 302-333, Open Access link.

It contains the results you need. Let me know if you need me to expand further.


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