Given that the inverse-gamma distribution is the one-dimensional version of the inverse-Wishart distribution, why will (philosophically speaking) an inverse-Wishart distribution that originally has more than one dimension not become the inverse-gamma distribution when marginalized over all but one of the dimensions?
I.e. if $\mathbf{\Psi}$ is a 2x2-matrix, and $\mathbf{X}\sim \mathcal{W}^{-1}(\mathbf{\Psi}, \nu)$, where $$ \mathcal{W}^{-1}(\mathbf{\Psi}, \nu) = \frac{|\mathbf{\Psi}|^{\nu/2}}{2^\nu\Gamma_p(\frac{\nu}{2})}|\mathbf{X}|^{-\frac{\nu+3}{2}}e^{-\frac{1}{2}tr({\mathbf{\Psi X}^{-1}})} $$ When marginalized over all but $\mathbf{X}_{11}$, you get $$ \mathbf{X}_{11} \sim \frac{\mathbf{\Psi}_{11}^{(\nu-1)/2}}{2^{\frac{\nu-1}{2}}\Gamma_1(\frac{\nu-1}{2})}\mathbf{X}_{11}^{-\frac{\nu+1}{2}}e^{-\frac{1}{2}{\mathbf{\Psi}_{11} \mathbf{X}_{11}^{-1}}} $$ whereas the inverse-gamma distribution for $\mathbf{X}_{11}$ with the corresponding parameters is $$ \mathbf{X}_{11} \sim \frac{\mathbf{\Psi}_{11}^{\nu/2}}{2^{\frac{\nu}{2}}\Gamma_1(\frac{\nu}{2})}\mathbf{X}_{11}^{-\frac{\nu+2}{2}}e^{-\frac{1}{2}{\mathbf{\Psi}_{11} \mathbf{X}_{11}^{-1}}}. $$
It seems to me that when you marginalize the 2x2 expression over all elements that are not $\mathbf{X}_{11}$, you should end up with a distribution that 'tells the same story': i.e., marginalizing over a parameter essentially means that we assume no knowledge of it - it might as well not exist. And that is the same assumption that gives rise to the inverse-gamma distribution, to my mind. So why do the two distributions not end up the same?