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I do know that inverse gamma is a conjugate prior for univariate normal distribution. I guess it's also a conjugate prior for multivariate normal distribution. I'm trying to get the closed-form posterior from an inverse gamma prior and a likelihood based on a multivariate normal distribution expecting to get inverse gamma posterior but I don't have any success yet. I appreciate it if you could help me get the posterior. Below is the model:

$$ Y_i | \kappa_i \sim \mathcal{N}_j(0, \kappa_i^2 1 1^t + \sigma^2 I_j)$$ $$ \kappa_i \sim \text{InverseGamma} (A, B)$$

Where $A$, $B$, and $\sigma^2$ are known constants. $1$ is a vector of 1 of length $j$ and $I$ is a $j$ by $j$ identity matrix.

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I believe that the problem is with your guess that the inverse gamma is so easily extended to the multivariate case.

From the distributions appendix of Gleman et al, Bayesian Data Analysis (3rd Edition), 582

The Inverse-Wishart distribution is the conjugate prior distribution for the multivariate normal co-variance matrix. ...

The Wishart distribution is the conjugate prior distribution for the inverse co-variance matrix in a multivariate normal distribution and is a multivariate generalization of the gamma distribution.

I'm uncertain whether you'd like to proceed in your own investigation with this hint, or if you'd like me to spill the beans and post a full solution. (Though, turning to page 73 of the same text, we find the particular underlying algebra that you're interested in.)

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