  We propose a set of causal estimands that we call ``the mediated
probabilities of causation.'' These estimands quantify the probabilities that
an observed negative outcome was induced via a mediating pathway versus a
direct pathway in a stylized setting involving a binary exposure or
intervention, a single binary mediator, and a binary outcome. We outline a set
of conditions sufficient to identify these effects given observed data, and
propose a doubly-robust projection based estimation strategy that allows for
the use of flexible non-parametric and machine learning methods for estimation.
We argue that these effects may be more relevant than the probability of
causation, particularly in settings where we observe both some negative outcome
and negative mediating event, and we wish to distinguish between settings where
the outcome was induced via the exposure inducing the mediator versus the
exposure inducing the outcome directly. We motivate our quantities of interest
by discussing applications to legal and medical questions of causal
attribution.
