I am trying to fit a multivariate Gaussian with a non-diagonal covariance matrix $\Sigma$ using nested sampling.
Usually, in other Bayesian analyses, we would use a Inverse Wishart or LKJ prior on the covariance matrix and compute the posterior using the analytical form of the prior.
However, in nested sampling, we cannot do this. We must specify the prior in terms of a transform from uniform uncorrelated samples. For instance, for a 1d normal prior, the prior transform would be the percent point function of the normal (the inverse of the CDF).
norm.ppf(uniform_samples) gives you samples from the prior.
Unless I am mistaken, there is no function to transform uniform samples into a Wishart or LKJ prior.
Is there a better prior for covariance matrices that has a usable transform? Or is there a technqiue to estimate these transforms? Using a uniform prior on covariance matrix elements seems foolish.