  We conduct non-asymptotic analysis on the mean-field variational inference
for approximating posterior distributions in complex Bayesian models that may
involve latent variables. We show that the mean-field approximation to the
posterior can be well-approximated relative to the Kullback-Leibler divergence
discrepancy measure by a normal distribution whose center is the maximum
likelihood estimator (MLE). In particular, our results imply that the center of
the mean-field approximation matches the MLE up to higher-order terms and there
is essentially no loss of efficiency in using it as a point estimator for the
parameter in any regular parametric model with latent variables. We also
propose a new class of variational weighted likelihood bootstrap (VWLB) methods
for quantifying the uncertainty in the mean-field variational inference. The
proposed VWLB can be viewed as a new sampling scheme that produces independent
samples for approximating the posterior. Comparing with traditional sampling
algorithms such Markov Chain Monte Carlo, VWLB can be implemented in parallel
and is free of tuning.
