I’m looking to accurately describe the density function of a multivariate posterior probability distribution based on samples from MCMC. As far as I know, in most cases this is done either with a simple parametric fit (e.g. fitting or updating a Gaussian distribution based on the posterior samples) or with kernel density estimates. But especially in higher dimensions a KDE is usually very poor; and parametric distributions may not fit the shape of the posterior very well.
The answer to this question mentions that much more efficient estimates may be available, at least in some cases. In my case I don’t have the conditional densities though, so I believe the methods for Gibbs sampling can’t be used. The book chapter ‘Estimating Marginal Posterior Densities‘ also only mentions KDE’s and methods that apply for Gibbs sampling, but not more general MCMC techniques (and they talk about marginal distributions whereas I would like to describe the joint).
I could imagine other general techniques for density estimation can be used (such as mixture modeling), but I would expect you can do better than this, especially when you do have a good estimate of the marginal likelihood as well. Am I missing something? Can anyone point me in the right direction?