It is alleged in this question and in the Wikipedia article and elsewhere that the density function for the inverse-Wishart distribution with $n$ degrees of freedom on $p\times p$ positive-definite symmetric matrices is $$ f(x) = c\times (\det x)^{-(n+p+1)/2} \exp\left( \frac{-1}2 \operatorname{tr} \left(\Sigma x^{-1}\right) \right)$$ where $\Sigma$ is the scale matrix.

With respect to what measure is this the density? One could guess that it means $$ \Pr( X\in A) = \int_A f(x) \prod_{i,j} dx_{ij} $$ where $x_{ij}$ is the $ij$ entry of the matrix $x,$ i.e. it's with respect to Lebesgue measure on a $p^2$-dimensional space. But then it's really a $p(p-1)/2$-dimensional space, since $x$ is symmetric, and that complicates the question. And that's something of a wild guess. The bounds of integration would be messy since the support of the density is the space of symmetric positive-definite matrices.


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