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This paper introduces a novel Bayesian framework, Bayesian Modeling of Threshold Manipulation via Mixtures (BMTM), and its hierarchical extension (HBMTM), for estimating causal effects under strategic customer manipulation of spending thresholds. The core idea is to model the observed spending distribution as a mixture of two latent distributions: one for customers strategically affected by the threshold (bunching) and another for those unaffected (non-bunching). The method employs a two-step Bayesian inference approach and extends to a hierarchical setting to estimate heterogeneous causal effects across subgroups. Empirically, the proposed methods, especially HBMTM, demonstrate more accurate and reliable estimates of causal effects compared to conventional Regression Discontinuity Design (RDD) methods in simulation studies and a real-world marketing dataset, even with small subgroup sample sizes.

The paper addresses several bottlenecks in causal inference when thresholds are involved:

Violation of RDD assumptions: Standard Regression Discontinuity Design (RDD) assumes local randomization around a threshold. This assumption is violated when customers, aware of the thresholds, strategically manipulate their behavior (e.g., spending) to qualify for rewards, breaking the continuity condition.

Novel Bayesian Mixture Model (BMTM): Proposes a new Bayesian framework to estimate causal effects under threshold manipulation by modeling the observed spending distribution as a mixture of two latent distributions: one for strategically affected (bunching) customers and one for unaffected (non-bunching) customers.
Two-step Bayesian Inference: Develops a robust two-step Bayesian approach to fit the mixture model, first modeling non-bunching customers and then fitting the mixture model to a sample around the threshold.
Hierarchical Bayesian Extension (HBMTM): Extends BMTM to a hierarchical Bayesian setting to estimate heterogeneous causal effects across customer subgroups, ensuring stable estimation even for small subgroup sample sizes by borrowing information.
Theoretical Justification: Demonstrates posterior contraction of the resulting posterior distribution of the causal effect under large samples, with a convergence rate of O(n⁻¹/²).
Empirical Validation: Demonstrates the effectiveness of BMTM and HBMTM through simulation studies and a real-world marketing dataset, showing superior performance over conventional RDD methods.

The paper proposes Bayesian Modeling of Threshold Manipulation via Mixtures (BMTM) and its hierarchical extension (HBMTM).

Causal Effect Definition The average treatment effect (ATT) of a threshold K on customers is defined as: Δ = E[Y(1) – Y(0) | Y(t) ∈ NK, t∈ {0,1}] where Y(1) is the potential outcome with threshold K, Y(0) is the potential outcome without a threshold, and NK = [y, y] is the neighborhood of the threshold K.

Mixture Model The observed spending distribution Yᵢ for customers within the neighborhood NK is modeled as a mixture of two distributions: p(Yᵢ|Ψ) = πf(Yᵢ|γ) + (1 − π)g(Yᵢ|θ) where Ψ = {π,θ,γ} are unknown parameters.

f(Yᵢ|γ) represents the distribution of "bunching" customers (strategically affected by the threshold).
g(Yᵢ|θ) represents the distribution of "non-bunching" customers (unaffected by the threshold).
π is the mixing parameter, representing the proportion of bunching customers.

The ATT can then be expressed as: Δ(Ψ) = (∫Y∈NK Yf(y|γ)dy / ∫Y∈NK f(y|γ)dy) - (∫Y∈NK Yg(y|θ)dy / ∫Y∈NK g(y|θ)dy)

Two-Step Bayesian Inference

Step 1 (Non-bunching distribution): Estimate the non-bunching distribution g(Yᵢ|θ) from observations Dkᶜ = {Yᵢ; Yᵢ ∈ Nkᶜ} (outside the neighborhood NK). A prior distribution π(θ) is assigned to θ, and the posterior p(θ|Dkᶜ) is obtained using an adjusted likelihood L(θ; Dkᶜ) to account for the probability of mass falling into NK.
Step 2 (Mixture Model): Fit the mixture model (p(Yᵢ|Ψ)) to observations DK = {Yᵢ; Yᵢ ∈ NK} (around the threshold) using the estimated g(Yᵢ|θ) from Step 1. A joint prior p(π, γ) is assigned to (π, γ), and the posterior p(π, γ|D) is computed by integrating over the posterior of θ from Step 1.

Hierarchical Bayesian Modeling of Threshold Manipulation (HBMTM) To estimate heterogeneous causal effects across G subgroups, the framework is extended hierarchically.

For each subgroup g, the generative distribution is: P_g(Y_gi|Ψ_g) = π_g f(Y_gi|γ_g) + (1 − π_g)g(Y_gi|θ_g).
Random effects structures are introduced for parameters: θ_g ~ H_θ(α_θ), γ_g ~ H_γ(α_γ), and logit(π_g/(1-π_g)) ~ N(μ_π, σ_π²).
The inference proceeds in two steps similar to BMTM, but for all groups simultaneously, borrowing strength across subgroups.

Bunching Distribution: For numerical examples, a skew-normal distribution (Azzalini 1985) is employed for f(y|γ) due to its flexibility in modeling a density peak slightly offset from the threshold.
Non-bunching Distribution: For numerical illustrations, a Singh-Maddala distribution (Singh and Maddala 1976) is adopted for g(y|θ) due to its flexibility and fit to income and expenditure data.
Latent Variable: The latent binary variable Zᵢ indicates bunching (Zᵢ=1) or non-bunching (Zᵢ=0), with P(Zᵢ=1) = π.
Posterior Computation: Markov Chain Monte Carlo (MCMC) algorithm using the probabilistic programming language Stan (Carpenter et al. 2017) is used for posterior computation, leveraging the Hamiltonian Monte Carlo algorithm (Duane et al. 1987).
Hierarchical Parameters:
- For g(Y_gi|θ_g) with θ_g = (a_g,b_g,q_g), log-normal random effects are assumed: log(a_g) ~ N(μ_a, σ_a²), log(b_g) ~ N(μ_b, σ_b²), log(q_g) ~ N(μ_q, σ_q²).
- For f(Y_gi|γ_g) with γ_g = (β_g,ω_g,δ_g), normal random effects are assumed: β_g ~ N(μ_β, σ_β²), log(ω_g) ~ N(μ_ω, σ_ω²), and δ_g ~ N(μ_δ, σ_δ²).
- For π_g, a logit-normal distribution is used: log{π_g/(1-π_g)} ~ N(μ_π, σ_π²).
Software: CmdStanPy (version 1.2.5) is used for model fitting.

Datasets:
- Simulation Studies: Data generated from a population structured into G = 100 distinct subgroups, with subgroup-specific sample sizes (n_g) set to (50, 100, 200, 300) within four clusters. Two scenarios were evaluated: (A) moderate bunching with low heterogeneity, and (B) sparse bunching with higher heterogeneity.
- Real-world Marketing Dataset: A month-long marketing promotion dataset where consumers could earn incentives by exceeding spending thresholds. Three thresholds were present: K₁ = 30,000 yen, K₂ = 50,000 yen, and K₃ = 70,000 yen. Subgroups (G = 21) were defined based on previous month's spending in 10,000 yen intervals.
Metrics:
- Point Estimation: Mean Absolute Error (MAE) = (1/G) Σ_g=1^G |Δ_g - Δ̂_g|
- Interval Estimation: Coverage Probability (CP), Average Length (AL), and Interval Score (IS). CP ideal value is 1-α (0.9 in this paper).
Baselines: Standard Regression Discontinuity Design (RDD) method.
Numeric Results (Simulation Studies - Table 1):
- Scenario (A):
  - RDD: MAE = 3.31
  - BMTM: MAE = 0.78, CP = 0.91, AL = 4.03, IS = 4.65
  - HBMTM: MAE = 0.33, CP = 0.84, AL = 1.20, IS = 1.85
- Scenario (B):
  - RDD: MAE = 3.50
  - BMTM: MAE = 1.79, CP = 0.94, AL = 9.05, IS = 9.88
  - HBMTM: MAE = 0.37, CP = 0.88, AL = 1.62, IS = 2.15
Key Findings:
- HBMTM consistently outperforms BMTM and RDD, achieving the lowest MAE and IS in both scenarios.
- In Scenario (B) with weaker bunching, BMTM's accuracy degrades (MAE from 0.78 to 1.79), while HBMTM remains stable and accurate (MAE 0.33 to 0.37).
- HBMTM maintains reliable CP across both scenarios.
- Real-world application shows significant heterogeneity in causal effects across subgroups, with positive effects for lower baseline spending and diminishing/negative effects for higher baseline spending.
Runtime/Training Cost: For simulated dataset, models were fitted based on 12,000 posterior samples (four chains, 3,000 samples each after burn-in). Not stated in the paper.

No explicit paper-reported limitations were found.

Method evidence: "To address causal inference under manipulation, we propose a novel Bayesian framework called Bayesian Modeling of Threshold Manipulation via Mixtures (BMTM). This framework employs a Bayesian mixture model to distinguish between customers strategically affected by a threshold and those who are not, enabling robust causal effect estimation with uncertainty quantification (see Figure 1). 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. In addition, we extend BMTM to a hierarchical Bayesian setting to estimate heterogeneous causal effects across subgroups of customers, which we call hierarchical BMTM (HBMTM)." (section: 1 Introduction)
Result evidence: "In both scenarios, our proposed methods, BMTM and HBMTM, substantially outperform the baseline RDD. In particular, HBMTM consistently demonstrates superior performance with the lowest MAE and IS, and the superiority is evident in Scenario (B). In Scenario (B) with a weaker bunching signal, the accuracy of BMTM degrades (MAE increases from 0.78 to 1.79), while that of HBMTM remains remarkably stable and accurate." (section: 5 Simulation Studies)
Limitation evidence: No explicit paper-reported limitations were found.

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

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