This manuscript enriches the framework of continuous normalizing flows (CNFs) within causal inference, primarily to augment the geometric properties of parametric submodels used in targeted maximum likelihood estimation (TMLE). By introducing an innovative application of CNFs, we construct a refined series of parametric submodels that enable a directed interpolation between the prior distribution $p_0$ and the empirical distribution $p_1$. This proposed methodology serves to optimize the semiparametric efficiency bound in causal inference by orchestrating CNFs to align with Wasserstein gradient flows. Our approach not only endeavors to minimize the mean squared error in the estimation but also imbues the estimators with geometric sophistication, thereby enhancing robustness against misspecification. This robustness is crucial, as it alleviates the dependence on the standard $n^{\frac{1}{4}}$ rate for a doubly-robust perturbation direction in TMLE. By incorporating robust optimization principles and differential geometry into the estimators, the developed geometry-aware CNFs represent a significant advancement in the pursuit of doubly robust causal inference.

翻译：本文丰富了因果推断中连续标准化流（CNFs）的框架，主要用于增强目标最大似然估计（TMLE）中参数子模型的几何性质。通过引入CNFs的创新应用，我们构建了一系列精细化的参数子模型，使得先验分布$p_0$和经验分布$p_1$之间能够进行定向插值。所提出的方法通过将CNFs与Wasserstein梯度流对齐，优化了因果推断中的半参数效率界。我们的方法不仅致力于最小化估计中的均方误差，还赋予估计量几何复杂性，从而增强其对抗错误设定的鲁棒性。这种鲁棒性至关重要，因为它减轻了在TMLE中双重鲁棒扰动方向对标准$n^{\frac{1}{4}}$率的依赖。通过将鲁棒优化原理和微分几何融入估计量，所开发的面向几何的CNFs代表了追求双重鲁棒因果推断的重要进展。