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We prove the main rules of causal calculus (also called do-calculus) for i/ostructural causal models (ioSCMs), a generalization of a recently proposedgeneral class of non-/linear structural causal models that allow for cycles,latent confounders and arbitrary probability distributions. We also generalizeadjustment criteria and formulas from the acyclic setting to the general one(i.e. ioSCMs). Such criteria then allow to estimate (conditional) causaleffects from observational data that was (partially) gathered under selectionbias and cycles. This generalizes the backdoor criterion, theselection-backdoor criterion and extensions of these to arbitrary ioSCMs.Together, our results thus enable causal reasoning in the presence of cycles,latent confounders and selection bias. Finally, we extend the ID algorithm forthe identification of causal effects to ioSCMs.

Causal Calculus in the Presence of Cycles, Latent Confounders and Selection Bias