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.
"Asymptotic normality and valid inference for Gaussian variational approximation." Ann. Statist. 39 (5) 2502 - 2532, October 2011. https://doi.org/10.1214/11-AOS908