  Causal inference from observational data requires assumptions. These
assumptions range from measuring confounders to identifying instruments.
Traditionally, causal inference assumptions have focused on estimation of
effects for a single treatment. In this work, we construct techniques for
estimation with multiple treatments in the presence of unobserved confounding.
We develop two assumptions based on shared confounding between treatments and
independence of treatments given the confounder. Together, these assumptions
lead to a confounder estimator regularized by mutual information. For this
estimator, we develop a tractable lower bound. To recover treatment effects, we
use the residual information in the treatments independent of the confounder.
We validate on simulations and an example from clinical medicine.
