Asymptotic normality and valid inference for Gaussian variational approximation

We derive the precise asymptotic distributional behavior of Gaussianvariational approximate estimators of the parameters in a single-predictorPoisson mixed model. These results are the deepest yet obtained concerningthe statistical properties of a variational approximation method. Moreover,they give rise to asymptotically valid statistical inference. A simulation studydemonstrates that Gaussian variational approximate confidence intervals possessgood to excellent coverage properties, and have a similar precision totheir exact likelihood counterparts.