  Graphical causal models led to the development of complete non-parametric
identification theory in arbitrary structured systems, and general approaches
to efficient inference. Nevertheless, graphical approaches to causal inference
have not been embraced by the statistics and public health communities. In
those communities causal assumptions are instead expressed in terms of
potential outcomes, or responses to hypothetical interventions. Such
interventions are generally conceptualized only on a limited set of variables,
where the corresponding experiment could, in principle, be performed. By
contrast, graphical approaches to causal inference generally assume
interventions on all variables are well defined - an overly restrictive and
unrealistic assumption that may have limited the adoption of these approaches
in applied work in statistics and public health. In this paper, we build on a
unification of graphical and potential outcomes approaches to causality
exemplified by Single World Intervention Graphs (SWIGs) to define graphical
models with a restricted set of allowed interventions. We give a complete
identification theory for such models, and develop a complete calculus of
interventions based on a generalization of the do-calculus, and axioms that
govern probabilistic operations on Markov kernels. A corollary of our results
is a complete identification theory for causal effects in another graphical
framework with a restricted set of interventions, the decision theoretic
graphical formulation of causality.
