  Continuously-indexed flows (CIFs) have recently achieved improvements over
baseline normalizing flows on a variety of density estimation tasks. CIFs do
not possess a closed-form marginal density, and so, unlike standard flows,
cannot be plugged in directly to a variational inference (VI) scheme in order
to produce a more expressive family of approximate posteriors. However, we show
here how CIFs can be used as part of an auxiliary VI scheme to formulate and
train expressive posterior approximations in a natural way. We exploit the
conditional independence structure of multi-layer CIFs to build the required
auxiliary inference models, which we show empirically yield low-variance
estimators of the model evidence. We then demonstrate the advantages of CIFs
over baseline flows in VI problems when the posterior distribution of interest
possesses a complicated topology, obtaining improved results in both the
Bayesian inference and surrogate maximum likelihood settings.
