Identifying causal effects in maximally oriented partially directed acyclic graphs

We develop a necessary and sufficient causal identification criterion for maximally oriented partially directed acyclic graphs (MPDAGs). MPDAGs as a class of graphs include directed acyclic graphs (DAGs), completed partially directed acyclic graphs (CPDAGs), and CPDAGs with added background knowledge. As such, they represent the type of graph that can be learned from observational data and background knowledge under the assumption of no latent variables. Our identification criterion can be seen as a generalization of the g-formula of Robins (1986). We further obtain a generalization of the truncated factorization formula (Pearl, 2009) and compare our criterion to the generalized adjustment criterion of Perkovi\'c et al. (2017) which is sufficient, but not necessary for causal identification.