  Causal mediation analysis with random interventions has become an area of
significant interest for understanding time-varying effects with longitudinal
and survival outcomes. To tackle causal and statistical challenges due to the
complex longitudinal data structure with time-varying confounders, competing
risks, and informative censoring, there exists a general desire to combine
machine learning techniques and semiparametric theory. In this manuscript, we
focus on targeted maximum likelihood estimation (TMLE) of longitudinal natural
direct and indirect effects defined with random interventions. The proposed
estimators are multiply robust, locally efficient, and directly estimate and
update the conditional densities that factorize data likelihoods. We utilize
the highly adaptive lasso (HAL) and projection representations to derive new
estimators (HAL-EIC) of the efficient influence curves of longitudinal
mediation problems and propose a fast one-step TMLE algorithm using HAL-EIC
while preserving the asymptotic properties. The proposed method can be
generalized for other longitudinal causal parameters that are smooth functions
of data likelihoods, and thereby provides a novel and flexible statistical
toolbox.
