  Unobserved confounding is one of the main challenges when estimating causal
effects. We propose a novel causal reduction method that replaces an arbitrary
number of possibly high-dimensional latent confounders with a single latent
confounder that lives in the same space as the treatment variable without
changing the observational and interventional distributions entailed by the
causal model. After the reduction, we parameterize the reduced causal model
using a flexible class of transformations, so-called normalizing flows. We
propose a learning algorithm to estimate the parameterized reduced model
jointly from observational and interventional data. This allows us to estimate
the causal effect in a principled way from combined data. We perform a series
of experiments on data simulated using nonlinear causal mechanisms and find
that we can often substantially reduce the number of interventional samples
when adding observational training samples without sacrificing accuracy. Thus,
adding observational data may help to more accurately estimate causal effects
even in the presence of unobserved confounders.
