On the Identifiability and Estimation of Functional Causal Models in the Presence of Outcome-Dependent Selection

We study the identifiability and estimation of functional causal models under selection bias, with a focus on the situation where the selection depends solely on the e↵ect variable, which is known as outcome-dependent selection. We address two questions of identifiability: the identifiability of the causal direction between two variables in the presence of selection bias, and, given the causal direction, the identifiability of the model with outcomedependent selection. Regarding the first, we show that in the framework of post-nonlinear causal models, once outcome-dependent selection is properly modeled, the causal direction between two variables is generically identifiable; regarding the second, we identify some mild conditions under which an additive noise causal model with outcome-dependent selection is to a large extent identifiable. We also propose two methods for estimating an additive noise model from data that are generated with outcome-dependent selection.