uncertainty quantification of mean field variational bayesian inference for gaussian mixture model

Mean field variational bayesian inference (mfvbi) is a computational tool in bayesian statistics to approximate posterior probability density p by a simpler one q, more easily accessible for doing inference and computation. to do this the basic idea is doing optimization through kulback leibler divergence d_kl (q||p). the tool has been successfully applied for many big data statistical models and it generalizes the standard mean field approximation in statistical mechanics for models in statistical learning. meanwhile it suffers from lack of mathematical uncertainty quantification. our goal in this paper is to answer this question for gaussian mixture models (gmm).