# Marginal distribution of the diagonal of an inverse Wishart distributed matrix

Suppose $X\sim InvWishart(\nu, \Sigma_0)$. I'm interested in the marginal distribution of the diagonal elements $diag(X) = (x_{11}, \dots, x_{pp})$. There are a few simple results on the distribution of submatrices of $X$ (at least some listed at Wikipedia). From this I can figure that the marginal distribution of any single element on the diagonal is inverse Gamma. But I've been unable to deduce the joint distribution.

I thought maybe it could be derived by composition, like

$p(x_{11} | x_{ii}, i\gt 1)p(x_{22}|x_{ii}, i>2)\dots p(x_{(p-1)(p-1)}|x_{pp})p(x_{pp})$

but I never got anywhere with it and further suspect that I'm missing something simple; it seems like this "ought" to be known but I haven't been able to find/show it.

-
Proposition 7.9 of Bilodeau and Brenner (the pdf is freely available on the web) gives a promising result for the Wishart (perhaps it carries over for the inverse Wishart). If you partition $X$ in blocks as $X_{11},X_{12};X_{21},X_{22}$, then $X_{22}$ is Wishart, as is $X_{11} - X_{12}X_{22}^{-1}X_{21}$, and they are independent. – shabbychef Oct 23 '11 at 4:34
That proposition only applies if you know the whole matrix: if you've only got the diagonal, then you don't know e.g. $X_{12}$, so you can't do the transformation. – petrelharp Sep 18 '13 at 21:09