  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 $\Gamma$-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.
