Establishing consistency and improving uncertainty estimates of variational inference through M-estimation

Variational inference has gained popularity over the past decade as a scalable estimation procedure for latent variable models with a simple and appealing intuition. In predictive settings, implementations using variational inference often have similar predictive performance to slower, exact alternatives. We know considerably less, however, about the viability of variational inference in contexts where parameter estimation and model interpretation are the primary goals. In this paper we connect variational inference to $M$-estimation, thereby providing general conditions for consistency and asymptotic normality of variational point estimators. We also derive a "sandwich" asymptotic covariance matrix that can be estimated using the output of a variational algorithm and is robust to model misspecification. We apply our methods to a particular type of variational inference known as Black Box Variational Inference and conduct a thorough simulation study demonstrating the validity of our derived covariance matrix under correct model specification as well as model misspecification.