Asymptotic normality and valid inference for Gaussian variational   approximation
  We derive the precise asymptotic distributional behavior of Gaussian variational approximate estimators of the parameters in a single-predictor Poisson mixed model. These results are the deepest yet obtained concerning the statistical properties of a variational approximation method. Moreover, they give rise to asymptotically valid statistical inference. A simulation study demonstrates that Gaussian variational approximate confidence intervals possess good to excellent coverage properties, and have a similar precision to their exact likelihood counterparts. 
Peter Hall, Tung Pham, M. P. Wand, S. S. J. Wang
math.ST stat.TH