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I'm currently working on a project around the Bayesian approach to portfolio selection, and I can't manage to wrap my mind around the specification of the non-informative (diffuse) prior. Assuming gaussianity of the returns with parameters $\mu \in R^N$ and $\Sigma \in R^{N x N}$ the prior is usually expressed as follows:

$p(\mu,\Sigma) \propto |\Sigma|^{-(N+1)/2}$

Where $|\Sigma|$ is $det(\Sigma)$ and $N$ the number of variables (assets in my case). I understand that this comes from the marginal priors of $\mu$ and $\Sigma$, given by

$p(\mu) \propto c$

$p(\Sigma) \propto |\Sigma|^{-(N+1)/2}$

I understand the specification of $p(\mu)$, so that's not a problem. However, I do not understand the specification of $p(\Sigma)$. I understand that in the special case $N=1$, we obtain the usual prior for scale parameters $p(\sigma) \propto 1/\sigma$, but I don't get how we obtain $|\Sigma|^{-(N+1)/2}$.

I've checked already well-known references: Zellner, 1971 / Bawa, Brown and Klein 1974 / Geisser 1965 / etc but it doesn't help. It seems that there is something that I fundamentally don't grasp.

If anyone has an intuitive and clear explanation for this, it would be great !!

Many thanks in advance.

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