Bayesian Statistics

Abstract Many marketing applications, including credit card incentive programs, offer rewards to customers exceeding specific spending thresholds to encourage increased consumption. Quantifying the causal effect of these thresholds on customers is crucial for effective marketing strategy design. While regression discontinuity design is a common 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 of thresholds on customers under their manipulation. The core 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 Bayesian approach, which enables valid causal effect estimation with proper uncertainty quantification. 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 show that our proposed framework yields more accurate estimates of the causal effect of thresholds on customers compared to naive regression discontinuity design methods.

Aug 4, 2025