  Variational inference is a general approach for approximating complex density
functions, such as those arising in latent variable models, popular in machine
learning. It has been applied to approximate the maximum likelihood estimator
and to carry out Bayesian inference, however, quantification of uncertainty
with variational inference remains challenging from both theoretical and
practical perspectives. This paper is concerned with developing uncertainty
measures for variational inference by using bootstrap procedures. We first
develop two general bootstrap approaches for assessing the uncertainty of a
variational estimate and the study the underlying bootstrap theory in both
fixed- and increasing-dimension settings. We then use the bootstrap approach
and our theoretical results in the context of mixed membership modeling with
multivariate binary data on functional disability from the National Long Term
Care Survey. We carry out a two-sample approach to test for changes in the
repeated measures of functional disability for the subset of individuals
present in 1989 and 1994 waves.
