We extend Robins' theory of causal inference for complex longitudinal data to the case of continuously varying as opposed to discrete covariates and treatments. In particular we establish versions of the key results of the discrete theory: the g-computation formula and a collection of powerful characterizations of the g-null hypothesis of no treatment effect. This is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome variable and of the covariates given the past. We also show that our assumptions concerning counterfactual variables place no restriction on the joint distribution of the observed variables: thus in a precise sense, these assumptions are "for free," or if you prefer, harmless.

翻译：我们将Robins关于复杂纵向数据因果推断的理论扩展至协变量和处理连续变化而非离散的情况。具体而言，我们建立了离散理论关键结果的版本：g-计算公式以及无处理效应g-零假设的一组强有力刻画。这是在关于结果变量和协变量给定历史的条件分布的自然连续性假设下完成的。我们还表明，我们关于反事实变量的假设对观测变量的联合分布不施加任何限制：因此在精确意义上，这些假设是“免费的”，或者如果你愿意，是无害的。