Abstract
We propose a parsimonious parameterization for causal models using a ``causal reduction'' technique. This technique: - shows that the observational distribution p(x,y) and the interventional distributions (p(y|do(x)) in Pearl's notation, {p(y(x))} in potential outcome notation) are not variationally independent objects; - allows to derive bounds on the (conditional) causal effect in terms of the observational distribution; - inspires estimators of the (conditional) causal effect that use a combination of observational and interventional data that do not suffer from bias due to unobserved confounding in the observed regime. We consider two cases in more detail. In the discrete case, we analyze maximum likelihood estimation and discuss simple settings in which the observational data is helpful for estimating the causal effect. In the continuous case we show empirically that non-parametric estimators based on normalizing flows may obtain more accurate estimates of causal effects from a combination of observational and interventional data than what could be obtained from the interventional data alone Joint work with Maximilian Ilse, Patrick Forré and Max Welling https://arxiv.org/abs/2103.04786v2