  Normalizing flows model a complex target distribution in terms of a bijective
transform operating on a simple base distribution. As such, they enable
tractable computation of a number of important statistical quantities,
particularly likelihoods and samples. Despite these appealing properties, the
computation of more complex inference tasks, such as the cumulative
distribution function (CDF) over a complex region (e.g., a polytope) remains
challenging. Traditional CDF approximations using Monte-Carlo techniques are
unbiased but have unbounded variance and low sample efficiency. Instead, we
build upon the diffeomorphic properties of normalizing flows and leverage the
divergence theorem to estimate the CDF over a closed region in target space in
terms of the flux across its \emph{boundary}, as induced by the normalizing
flow. We describe both deterministic and stochastic instances of this
estimator: while the deterministic variant iteratively improves the estimate by
strategically subdividing the boundary, the stochastic variant provides
unbiased estimates. Our experiments on popular flow architectures and UCI
benchmark datasets show a marked improvement in sample efficiency as compared
to traditional estimators.
