Establishing consistency and improving uncertainty estimates of variational inference through M-estimation
Ted Westling, Tyler H. McCormick · arXiv (Cornell University) · 2015
Variational inference (VI) is gaining popularity as a scalable estimation procedure for latent variable models. VI often empirically achieves similar predictive performance to slower, exact alternatives, but less is known about the viability of VI in contexts where parameter estimation and model interpretation are the primary goals. In this paper we connect VI for independent, identically distributed (IID) mixture models to M-estimation. We leverage extensive results about M-estimators from statistical theory to provide general conditions for consistency and asymptotic normality of VI point estimators. We also derive a sandwich asymptotic covariance matrix and a consistent estimator thereof. Our estimated covariance can be used to construct valid confidence regions and tests and is robust to model misspecification. We provide more specific conditions for the Gaussian Variational Approximation (GVA), which has been implemented in broad generality in the open-source software Stan. We conduct a thorough simulation study demonstrating our derived covariance matrix under correct and misspecified models and apply our methods to estimate a mixed effects logistic regression using data from the National Longitudinal Study of Adolescent Health.