标题： 阈值操纵下的因果推断：贝叶斯混合建模与异质处理效应 显示英文标题

标题： Causal Inference under Threshold Manipulation: Bayesian Mixture Modeling and Heterogeneous Treatment Effects

摘要： 许多营销应用，包括信用卡激励计划，向超过特定消费门槛的客户提供奖励，以鼓励增加消费。 量化这些门槛对客户的影响对于有效的营销策略设计至关重要。 尽管断点回归设计是此类因果推断任务的标准方法，但当客户了解门槛时，可能会战略性地调整其消费以符合奖励条件，从而违反其假设。 为解决这一问题，我们提出了一种新的框架，用于在门槛操纵下估计因果效应。 主要思想是将观察到的消费分布建模为两个分布的混合：一个代表受门槛战略影响的客户，另一个代表不受影响的客户。 为了拟合混合模型，我们采用了一个两步贝叶斯方法，包括对非聚集客户进行建模，并在门槛附近的样本上拟合混合模型。 我们在大样本下展示了因果效应后验分布的后验收缩。 此外，我们将该框架扩展到分层贝叶斯设置，以估计不同客户子群体之间的异质因果效应，即使在子群体样本量较小时也能实现稳定的推断。 我们通过模拟研究证明了所提出方法的有效性，并使用一个实际的营销数据集说明了它们的实际意义。 显示英文摘要

摘要： Many marketing applications, including credit card incentive programs, offer rewards to customers who exceed specific spending thresholds to encourage increased consumption. Quantifying the causal effect of these thresholds on customers is crucial for effective marketing strategy design. Although regression discontinuity design is a standard method for such causal inference tasks, its assumptions can be violated when customers, aware of the thresholds, strategically manipulate their spending to qualify for the rewards. To address this issue, we propose a novel framework for estimating the causal effect under threshold manipulation. The main idea is to model the observed spending distribution as a mixture of two distributions: one representing customers strategically affected by the threshold, and the other representing those unaffected. To fit the mixture model, we adopt a two-step Bayesian approach consisting of modeling non-bunching customers and fitting a mixture model to a sample around the threshold. We show posterior contraction of the resulting posterior distribution of the causal effect under large samples. Furthermore, we extend this framework to a hierarchical Bayesian setting to estimate heterogeneous causal effects across customer subgroups, allowing for stable inference even with small subgroup sample sizes. We demonstrate the effectiveness of our proposed methods through simulation studies and illustrate their practical implications using a real-world marketing dataset.