Design-based inference, also known as randomization-based or finite-population inference, provides a principled framework for trustworthy statistical inference by attributing randomness solely to the design mechanism (e.g., treatment assignment, survey sampling, or missingness), without imposing distributional or modeling assumptions on outcome data. Despite its conceptual appeal and long history, applying this framework becomes challenging when the underlying design probabilities (i.e., propensity scores) are unknown, as is common in observational studies, real-world surveys, and missing-data settings. Existing plug-in and matching-based methods either ignore uncertainty from propensity score estimation or rely on near-exact covariate matching, often leading to systematic under-coverage, while existing finite-population M-estimation approaches remain largely restricted to parametric propensity score models. In this work, we propose propensity score propagation, a general framework for valid design-based inference with unknown propensity scores. The framework introduces a regeneration-and-union procedure to propagate uncertainty from propensity score estimation into downstream design-based inference. It accommodates both parametric and nonparametric propensity score models, integrates seamlessly with existing design-based inference methods developed under known propensity scores, and applies broadly across design-based inference problems. Theoretical and simulation studies show that the proposed framework achieves nominal coverage, even in settings where conventional approaches exhibit substantial under-coverage.

翻译：基于设计的推断（也称为随机化推断或有限总体推断）通过将随机性完全归因于设计机制（如处理分配、调查抽样或数据缺失），不对结果数据施加分布或模型假设，为可信统计推断提供了严谨框架。尽管该框架在概念上具有吸引力且历史悠久，但当底层设计概率（即倾向性评分）未知时（常见于观察性研究、真实世界调查和缺失数据场景），应用该框架面临挑战。现有插件法和匹配法要么忽略倾向性评分估计的不确定性，要么依赖近乎精确的协变量匹配，常导致系统性覆盖不足；而现有有限总体M估计方法仍主要局限于参数化倾向性评分模型。本文提出倾向性评分传播框架，为未知倾向评分下的有效基于设计推断提供统一方案。该框架引入再生-联合程序，将倾向性评分估计的不确定性传播至下游基于设计的推断中。它兼容参数化与非参数化倾向性评分模型，可无缝集成现有已知倾向评分下的基于设计推断方法，并广泛适用于各类基于设计的推断问题。理论与模拟研究表明：即使在传统方法呈现显著覆盖不足的场景中，所提框架仍能达到名义覆盖水平。