  Given an inverse problem with a normalizing flow prior, we wish to estimate
the distribution of the underlying signal conditioned on the observations. We
approach this problem as a task of conditional inference on the pre-trained
unconditional flow model. We first establish that this is computationally hard
for a large class of flow models. Motivated by this, we propose a framework for
approximate inference that estimates the target conditional as a composition of
two flow models. This formulation leads to a stable variational inference
training procedure that avoids adversarial training. Our method is evaluated on
a variety of inverse problems and is shown to produce high-quality samples with
uncertainty quantification. We further demonstrate that our approach can be
amortized for zero-shot inference.
