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 singlepredictor 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 with precision similar to their exact likelihood counterparts.