On the Validity of Covariate Adjustment for Estimating Causal Effects
  Identifying effects of actions (treatments) on outcome variables from observational data and causal assumptions is a fundamental problem in causal inference. This identification is made difficult by the presence of confounders which can be related to both treatment and outcome variables. Confounders are often handled, both in theory and in practice, by adjusting for covariates, in other words considering outcomes conditioned on treatment and covariate values, weighed by probability of observing those covariate values. In this paper, we give a complete graphical criterion for covariate adjustment, which we term the adjustment criterion, and derive some interesting corollaries of the completeness of this criterion. 
Ilya Shpitser, Tyler VanderWeele, James M. Robins
stat.ME cs.AI