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Variational inference is a popular method for estimating model parameters andconditional distributions in hierarchical and mixed models, which arisefrequently in many settings in the health, social, and biological sciences.Variational inference in a frequentist context works by approximatingintractable conditional distributions with a tractable family and optimizingthe resulting lower bound on the log-likelihood. The variational objectivefunction is typically less computationally intensive to optimize than the truelikelihood, enabling scientists to fit rich models even with extremely largedatasets. Despite widespread use, little is known about the general theoreticalproperties of estimators arising from variational approximations to thelog-likelihood, which hinders their use in inferential statistics. In thispaper we connect such estimators to profile M-estimation, which enables us toprovide regularity conditions for consistency and asymptotic normality ofvariational estimators. Our theory also motivates three methodologicalimprovements to variational inference: estimation of the asymptoticmodel-robust covariance matrix, a one-step correction that improves estimatorefficiency, and an empirical assessment of consistency. We evaluate theproposed results using simulation studies and data on marijuana use from theNational Longitudinal Study of Youth.

Beyond prediction: A framework for inference with variational approximations in mixture models