In causal inference, an important problem is to quantify the effects of interventions or treatments. Many studies focus on estimating the mean causal effects; however, these estimands may offer limited insight since two distributions can share the same mean yet exhibit significant differences. Examining the causal effects from a distributional perspective provides a more thorough understanding. In this paper, we employ a semiparametric density ratio model (DRM) to characterize the counterfactual distributions, introducing a framework that assumes a latent structure shared by these distributions. Our model offers flexibility by avoiding strict parametric assumptions on the counterfactual distributions. Specifically, the DRM incorporates a nonparametric component that can be estimated through the method of empirical likelihood (EL), using the data from all the groups stemming from multiple interventions. Consequently, the EL-DRM framework enables inference of the counterfactual distribution functions and their functionals, facilitating direct and transparent causal inference from a distributional perspective. Numerical studies on both synthetic and real-world data validate the effectiveness of our approach.

翻译：在因果推断中，一个核心问题在于量化干预或处理的效应。许多研究聚焦于估计平均因果效应；然而，由于两个分布可能具有相同的均值却展现出显著差异，这些估计量所能提供的洞见可能有限。从分布视角审视因果效应能够提供更全面的理解。本文采用半参数密度比模型（DRM）来刻画反事实分布，引入了一个假设这些分布共享潜在结构的框架。我们的模型通过避免对反事实分布施加严格的参数化假设，提供了灵活性。具体而言，DRM包含一个非参数分量，该分量可通过经验似然（EL）方法，利用源自多重干预的所有组别数据进行估计。因此，EL-DRM框架能够推断反事实分布函数及其泛函，从而促进从分布视角进行直接且透明的因果推断。在合成数据与真实数据上的数值研究验证了我们方法的有效性。