Say I define a probability distribution on $P$ dimensional symmetric matrices such that the diagonals are strictly positive but the off diagonals are unrestricted (except for the symmetry constraint). Does this induce a proper probability distribution on the space of symmetric positive definite matrices?
A simple example like what I'm thinking about: $\sigma_{ii}\sim TN(0, 1)$ ($TN$ is the normal distribution truncated to be positive) and $\sigma_{ij}\sim N(0, 1)$ independently with $\sigma_{ji}=\sigma_{ij}$ so that $\pi(\Sigma) \propto [\prod_{p=1}^P N(0, 1) 1(\sigma_{ii}>0)][ \prod_{i
So my question is if $A$ is the set of SPD matrices, is $\pi_0(\Sigma) \propto \pi(\Sigma)1(\Sigma \in A)$ a probability distribution? Does this hold generally, or does it depend on the distributions for $\sigma_{ii}, \sigma_{ij}$ (assuming that they are all continuous), or on the size of $P$?
Leaving aside for the moment if this is actually useful or a good idea, does it work? To me it seems like...maybe. Intuitively it seems that it should, but I've learned not to trust my intuition. And I'm not sure about a good way to demonstrate that it does or doesn't, either.