  Mediation analysis is an important analytic tool commonly used in a broad
range of scientific applications. In this article, we study the problem of
mediation analysis when there are multivariate and conditionally dependent
mediators, and when the variables are observed over multiple time points. The
problem is challenging, because the effect of a mediator involves not only the
path from the treatment to this mediator itself at the current time point, but
also all possible paths pointed to this mediator from its upstream mediators,
as well as the carryover effects from all previous time points. We propose a
novel multivariate dynamic mediation analysis approach. Drawing inspiration
from the Markov decision process model that is frequently employed in
reinforcement learning, we introduce a Markov mediation process paired with a
system of time-varying linear structural equation models to formulate the
problem. We then formally define the individual mediation effect, built upon
the idea of simultaneous interventions and intervention calculus. We next
derive the closed-form expression and propose an iterative estimation procedure
under the Markov mediation process model. We study both the asymptotic property
and the empirical performance of the proposed estimator, and further illustrate
our method with a mobile health application.
