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 effect in practice. We therefore assume that the true treatment effect lies somewhere in between no treatment effect and the observational estimate, or in their convex hull. 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 Optimal 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.
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