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：本文在因果推断框架内丰富了连续归一化流（CNFs）的应用，主要目标是增强目标最大似然估计（TMLE）中参数子模型的几何特性。通过引入CNFs的创新性应用，我们构建了一系列精细化的参数子模型，实现了先验分布$p_0$与经验分布$p_1$之间的定向插值。该提出的方法通过调控CNFs使其与Wasserstein梯度流对齐，从而优化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，还为估计量注入了几何复杂性，从而增强了对错误设定的鲁棒性。这种鲁棒性至关重要，因为它降低了对TMLE中双鲁棒扰动方向所需标准$n^{\frac{1}{4}}$速率的依赖性。通过将鲁棒优化原理和微分几何融入估计量，所开发的几何感知CNFs代表了在追求双鲁棒因果推断方面的一项重要进展。