Given a sample $S = \{x_1,...,x_m\}$ from a multivariate Gaussian in ${n}$ dimensions, and the improper uniform prior over all possible such Gaussians, what is the posterior compound distribution?
In other words: Start with the improper uniform prior over all multivariate Gaussians in ${n}$ dimensions. Condition that prior on ${S}$, and find the corresponding posterior distribution ${P}$. Find the compound Gaussian distribution with mean and variance distributed according to ${P}$.
By improper prior over all possible Gaussians, I mean the prior that attributes the same likelihood to all means, and to all covariances.
Note that the final compound distribution will only be a proper distribution when ${m}$ is large enough (when ${n = 1}$, I believe ${m}$ needs to be at least 3).
(Partial answers will also be considered.)
The solution is a multivariate Student distribution of dimension $n$:
\begin{align*} \frac{\Gamma \left(\frac{m+n}{2}-1\right)}{\Gamma \left(\frac{m}{2}-1\right)}\frac{1}{\pi ^{n/2} (m+1)^{n/2}|\boldsymbol{\Sigma} |^{\frac{1}{2}}}\left(1+\frac{1}{(m+1)}(\mathbf{x}-\boldsymbol{\mu} )^T\boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu} )\right)^{-\left(\frac{m+n}{2}-1\right)} \end{align*} where $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$ are the sample mean vector and the sample covaraince matrix, respectively: \begin{align*} \boldsymbol{\mu} = \frac{1}{m} \sum_{k=1}^m \mathbf{x}_k, \quad \boldsymbol{\Sigma} = \frac{1}{m} \sum_{k=1}^m \mathbf{x}_k \mathbf{x}^{T}_k -\boldsymbol{\mu} \boldsymbol{\mu}^T. \end{align*}
As in the univariate case , we need $m>2$ for the distribution to be proper. Furthermore, to avoid degeneracy, the sample size must be larger than the dimension count: $m>n$.