  Causal intervention is an essential tool in causal inference. It is
axiomatized under the rules of do-calculus in the case of structure causal
models. We provide simple axiomatizations for families of probability
distributions to be different types of interventional distributions. Our
axiomatizations neatly lead to a simple and clear theory of causality that has
several advantages: it does not need to make use of any modeling assumptions
such as those imposed by structural causal models; it only relies on
interventions on single variables; it includes most cases with latent variables
and causal cycles; and more importantly, it does not assume the existence of an
underlying true causal graph as we do not take it as the primitive object--in
fact, a causal graph is derived as a by-product of our theory. We show that,
under our axiomatizations, the intervened distributions are Markovian to the
defined intervened causal graphs, and an observed joint probability
distribution is Markovian to the obtained causal graph; these results are
consistent with the case of structural causal models, and as a result, the
existing theory of causal inference applies. We also show that a large class of
natural structural causal models satisfy the theory presented here. We note
that the aim of this paper is axiomatization of interventional families, which
is subtly different from "causal modeling."
