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拓展了连续归一化流（CNFs）的理论体系，核心目标是通过增强靶向最大似然估计（TMLE）中参数子模型的几何特性实现方法创新。通过引入CNFs的创新性应用，我们构建了一个精炼的参数子模型序列，该序列能够实现先验分布$p_0$与经验分布$p_1$之间的定向插值。所提出的方法论通过将CNFs与Wasserstein梯度流对齐，优化了因果推断中的半参数效率界。该方法不仅致力于最小化估计量的均方误差，更赋予估计量几何精密性，从而增强对模型误设定的稳健性。这种稳健性至关重要，因为它降低了对TMLE中双重稳健扰动方向所需的标准$n^{\frac{1}{4}}$收敛速率的依赖。通过将稳健优化原理与微分几何融入估计量构建，本文开发的几何感知CNFs标志着双重稳健因果推断领域的重要进展。