Quantifying intrinsic causal contributions via structure preserving interventions

We propose a new notion of causal contribution which describes the 'intrinsic' part of the contribution of a node on a target node in a DAG. We show that in some scenarios the existing causal quantification methods failed to capture this notion exactly. By recursively writing each node as a function of the upstream noise terms, we separate the intrinsic information added by each node from the one obtained from its ancestors. To interpret the intrinsic 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 do not perturb the observed joint distribution. To get a measure that is invariant across arbitrary orderings of nodes we propose Shapley based symmetrization. We describe our contribution analysis for variance and entropy, but contributions for other target metrics can be defined analogously.