Construct and justify confidence intervals for long-term causality parameters estimated by machine learning. Longitudinal parameters include long-term, dynamic, and mediation effects. We provide a non-asymptotic theorem for a general class of longitudinal causal parameters estimated by any machine learning algorithm satisfying some simple conditions. Main results include local parameters defined for specific demographics and proximal parameters defined when unobserved confounding is present. We prove consistency, Gaussian approximation, and semiparametric efficiency. The Gaussian approximation rate is $n^{-1/2}$ for global parameters and falls off nicely for local parameters. We define a simple set of conditions for transforming the mean square rate into statistical inference and see that they hold for adversarial estimators in the general function space. A key feature of the main results is the novel multiple robustness against inappropriate postures for proximal causal inference in the long-term setting. As an independent interest, we provide what appears to be the first mean square rate for nested nonparametric instrumental variable regression.
