Quantifying causal contributions via structure preserving interventions

We introduce a concept to quantify the 'intrinsic' causal contribution of each variable in a causal directed acyclic graph to the uncertainty or information of some target variable. By recursively writing each node as function of the noise terms, we separate the information added by each node from the one obtained from its ancestors. To interpret this information as a causal contribution, we consider 'structure-preserving interventions' that randomize each node in a way that mimics the usual dependence on the parents and don't perturb the observed joint distribution. Using Shapley values, the contribution becomes independent of the ordering of nodes. We describe our contribution analysis for variance and entropy as two important examples, but contributions for other target metrics can be defined analogously.