Systematically comparing VI and MCMC results

I am trying to do some experiments with Variational Inference. In the process, I want to make sure that the posterior of VI is consistent with MCMC. Currently, I take the samples from both VI and MCMC and make a density plot to check how close VI gets to MCMC. Is there a more standard or statistical way of doing this comparison?

P.S.: I have searched along these lines for the already open threads but please guide me to one if you find any.

there are all kinds of things you can do. comparing univariate statistics (posterior means, posterior variances) and computing the ks statistic being some of the simpler ones.

note that unless you’re looking at pretty simple problems i wouldn’t expect VI and MCMC to match very closely.