Jeffreys prior of a multivariate Gaussian I have found two different expressions for the Jeffreys prior of a multivariate Gaussian. Eq. (3) in this article states that $$p(\mu,\Sigma) \propto \det(\Sigma)^{-(d+2)/2}$$
However in page 73 of this book it is claimed that $$p(\mu,\Sigma) \propto \det(\Sigma)^{-(d+1)/2}$$
Furthermore, in the first article it says that the proposed density

[...] is the only density which makes the Fisher information for the parameters invariant to all possible reparameterizations of the Gaussian.

Wouldn't this mean that the second prior cannot be Jeffreys, by definition? Is the second one wrong then?

Gelman is not wrong. One needs to understand the context.
For Inverse-Wishart distribution $\mathcal{IW}(\mathbf X;\mathbf \Psi,\nu),$ the relevant term of the density is $$ |\mathbf X|^{-\frac{(\nu+d+1)}2}\exp\left\{-\frac12\operatorname{tr}\left(\mathbf \Psi\mathbf X^{-1}\right)\right\}\tag 1\label 1$$ where $d$ is the order of $\mathbf X, ~\mathbf \Psi.$
The author of the paper takes $\mathcal{IW}(\mathbf V;k\mathbf V_0,k)$ and so $\eqref 1$ becomes as $k\to 0,$
$$ |\mathbf V|^{-\frac{(d+1)}2};$$ therefore $$p(\mathbf m, \mathbf V) \propto |\mathbf V|^{-\frac12}|\mathbf V|^{-\frac{(d+1)}2}=|\mathbf V|^{-\left(\frac{d}2+1\right)}.\tag 2$$
Gelman whereas considers $\mathcal{IW}\left(\mathbf \Sigma;\mathbf \Lambda_0, \nu_0\right)  $ and then takes $k_0\to 0,~ \nu_0\to-1, ~\det(\mathbf \Lambda_0) \to 0,$ (note previously $k$ was tending to $0$ but here its counterpart $\nu_0$ is tending to $-1$) to get
$$p(\boldsymbol\mu, \mathbf \Sigma)\propto |\mathbf\Sigma|^{-\frac{(d+1)}2}.\tag 3$$
In fact, as this paper notes, both are valid objective priors, the latter being the independence-Jeffreys prior.
