Keywords: observational study, positivity, stochastic intervention, time-varying confounding, treatment effect
Most work in causal inference considers deterministic interventions that set treatments to a fixed value. However, under positivity violations this can lead to non-identification, inefficiency, and effects with little practical relevance. Further, in longitudinal studies these effects are sensitive to the curse of dimensionality, resulting in widespread use of unrealistic parametric models. We propose a novel solution to these problems: incremental interventions that shift propensity score values. These interventions have several crucial advantages. First, they avoid positivity assumptions entirely. Second, they require no parametric assumptions but still admit a simple characterization of longitudinal effects, regardless of study length (effects are described with a curve instead of coefficients). After characterizing incremental interventions and giving identifying conditions for their effects, we develop general efficiency theory, propose efficient nonparametric estimators that can attain fast convergence rates even when incorporating flexible machine learning, and propose a bootstrap-based confidence band and simultaneous test of no treatment effect.