  What types of differences among causal structures with latent variables are
impossible to distinguish by statistical data obtained by probing each visible
variable? If the probing scheme is simply passive observation, then it is
well-known that many different causal structures can realize the same joint
probability distributions. Even for the simplest case of two visible variables,
for instance, one cannot distinguish between one variable being a causal parent
of the other and the two variables sharing a latent common cause. However, it
is possible to distinguish between these two causal structures if we have
recourse to more powerful probing schemes, such as the possibility of
intervening on one of the variables and observing the other. Herein, we address
the question of which causal structures remain indistinguishable even given the
most informative types of probing schemes on the visible variables. We find
that two causal structures remain indistinguishable if and only if they are
both associated with the same mDAG structure (as defined by Evans (2016)). We
also consider the question of when one causal structure dominates another in
the sense that it can realize all of the joint probability distributions that
can be realized by the other using a given probing scheme. (Equivalence of
causal structures is the special case of mutual dominance.) Finally, we
investigate to what extent one can weaken the probing schemes implemented on
the visible variables and still have the same discrimination power as a
maximally informative probing scheme.
