  Randomized clinical trials eliminate confounding but impose strict exclusion
criteria that limit recruitment to a subset of the population. Observational
datasets are more inclusive but suffer from confounding -- often providing
overly optimistic estimates of treatment response over time due to partially
optimized physician prescribing patterns. We therefore assume that the
unconfounded treatment response lies somewhere in-between the observational
estimate before and the observational estimate after treatment assignment. This
assumption allows us to extrapolate results from exclusive trials to the
broader population by analyzing observational and trial data simultaneously
using an algorithm called Optimum in Convex Hulls (OCH). OCH represents the
treatment effect either in terms of convex hulls of conditional expectations or
convex hulls (also known as mixtures) of conditional densities. The algorithm
first learns the component expectations or densities using the observational
data and then learns the linear mixing coefficients using trial data in order
to approximate the true treatment effect; theory importantly explains why this
linear combination should hold. OCH estimates the treatment effect in terms
both expectations and densities with state of the art accuracy.
