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Abstract:
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Variational inference (VI) has gained popularity over the past decade as a scalable estimation procedure for latent variable models with a simple and appealing intuition. 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 clustered, correlated, and longitudinal mixture models to M-estimation, thereby providing general conditions for consistency and asymptotic normality of VI point estimators. This connection also inspires two methodological improvements to VI. First, we can consistently estimate a "sandwich" asymptotic covariance matrix that vastly improves parameter uncertainty estimates and is robust to model misspecification. Second, if the marginal likelihood and its gradient can be tractably computed then a one-step correction makes the variational estimates asymptotically efficient. We apply our methods to logistic GLMMs with random intercepts and random slopes on simulated and real data.
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