Frequentist Consistency of Generalized Variational Inference
This provides theoretical guarantees for GVI methods, addressing a foundational issue in variational inference for statisticians and machine learning practitioners, though it is incremental as it builds on existing GVI frameworks.
The paper tackles the problem of establishing Frequentist consistency for Generalized Variational Inference (GVI) posteriors, showing that under minimal regularity conditions, these posteriors collapse to a point mass at the population-optimal parameter value as observations increase, with results extending to latent variables and misspecification.
This paper investigates Frequentist consistency properties of the posterior distributions constructed via Generalized Variational Inference (GVI). A number of generic and novel strategies are given for proving consistency, relying on the theory of $Γ$-convergence. Specifically, this paper shows that under minimal regularity conditions, the sequence of GVI posteriors is consistent and collapses to a point mass at the population-optimal parameter value as the number of observations goes to infinity. The results extend to the latent variable case without additional assumptions and hold under misspecification. Lastly, the paper explains how to apply the results to a selection of GVI posteriors with especially popular variational families. For example, consistency is established for GVI methods using the mean field normal variational family, normal mixtures, Gaussian process variational families as well as neural networks indexing a normal (mixture) distribution.