  Judea Pearl's insight that, when errors are assumed independent, the Pure
(aka Natural) Direct Effect (PDE) is non-parametrically identified via the
Mediation Formula was `path-breaking' in more than one sense! In the same paper
Pearl described a thought-experiment as a way to motivate the PDE. Analysis of
this experiment led Robins \& Richardson to a novel way of conceptualizing
direct effects in terms of interventions on an expanded graph in which
treatment is decomposed into multiple separable components. We further develop
this novel theory here, showing that it provides a self-contained framework for
discussing mediation without reference to cross-world (nested) counterfactuals
or interventions on the mediator. The theory preserves the dictum `no causation
without manipulation' and makes questions of mediation empirically testable in
future Randomized Controlled Trials. Even so, we prove the interventionist and
nested counterfactual approaches remain tightly coupled under a Non-Parametric
Structural Equation Model except in the presence of a `recanting witness.' In
fact, our analysis also leads to a simple sound and complete algorithm for
determining identification in the (non-interventionist) theory of path-specific
counterfactuals.
