  Mediation analysis in causal inference has traditionally focused on binary
exposures and deterministic interventions, and a decomposition of the average
treatment effect in terms of direct and indirect effects. In this paper we
present an analogous decomposition of the \textit{population intervention
effect}, defined through stochastic interventions on the exposure. Population
intervention effects provide a generalized framework in which a variety of
interesting causal contrasts can be defined, including effects for continuous
and categorical exposures. We show that identification of direct and indirect
effects for the population intervention effect requires weaker assumptions than
its average treatment effect counterpart, under the assumption of no
mediator-outcome confounders affected by exposure. In particular,
identification of direct effects is guaranteed in experiments that randomize
the exposure and the mediator. We discuss various estimators of the direct and
indirect effects, including substitution, re-weighted, and efficient estimators
based on flexible regression techniques, allowing for multivariate mediators.
Our efficient estimator is asymptotically linear under a condition requiring
$n^{1/4}$-consistency of certain regression functions. We perform a simulation
study in which we assess the finite-sample properties of our proposed
estimators. We present the results of an illustrative study where we assess the
effect of participation in a sports team on BMI among children, using mediators
such as exercise habits, daily consumption of snacks, and overweight status.
