Bayesian Fuzzy Regression Discontinuity
Also known as: Bayesian Fuzzy RD, Bayesian Fuzzy RDD, Fuzzy RD with Bayesian Inference
Bayesian Fuzzy Regression Discontinuity (Bayesian Fuzzy RD) combines the quasi-experimental logic of fuzzy regression discontinuity design with full Bayesian inference. It estimates a local average treatment effect at a policy threshold where treatment assignment is probabilistic rather than deterministic, placing prior distributions over all unknowns and recovering a complete posterior distribution of the causal effect rather than a single point estimate.
Key highlights
- Recovers a full posterior distribution of the LATE, enabling probability statements about the treatment effect rather than a binary reject-or-not decision.
- Performs well in small samples near the threshold where classical asymptotic confidence intervals are poorly calibrated.
- Handles imperfect compliance naturally through a joint structural model rather than requiring a two-stage least squares approximation.
- Incorporates prior information from domain experts or pilot studies, which can stabilise estimates when data near the cut-off are sparse.
- Produces credible intervals whose width directly reflects both sampling variation and model uncertainty.
Intuition
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How it works
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When to use it
Use Bayesian Fuzzy RD when a policy or programme assigns eligibility by crossing a numerical threshold but compliance is imperfect — meaning not everyone above the cut-off takes up treatment and some below it receive it regardless. It is especially suited when the sample near the threshold is small (fewer than a few hundred observations close to the cut-off), making asymptotic inference unreliable, and when quantifying full uncertainty in the LATE is scientifically important. It is not appropriate when compliance is perfect (use sharp Bayesian RD instead), when there is no credible discontinuity in take-up at the threshold, when the running variable is discrete with very few mass points near the cut-off, or when you cannot justify any prior distribution for the outcome surface.
Strengths & limitations
- Recovers a full posterior distribution of the LATE, enabling probability statements about the treatment effect rather than a binary reject-or-not decision.
- Performs well in small samples near the threshold where classical asymptotic confidence intervals are poorly calibrated.
- Handles imperfect compliance naturally through a joint structural model rather than requiring a two-stage least squares approximation.
- Incorporates prior information from domain experts or pilot studies, which can stabilise estimates when data near the cut-off are sparse.
- Produces credible intervals whose width directly reflects both sampling variation and model uncertainty.
- Results can be sensitive to the choice of prior for the outcome surface and compliance model, requiring thorough prior-sensitivity analysis.
- MCMC estimation is computationally demanding compared with classical local polynomial RD estimators.
- The LATE is local — it applies only to compliers at the threshold and cannot be extrapolated to units far from the cut-off.
- Requires careful bandwidth and kernel specification even in the Bayesian framework, and there is less consensus on automatic selection procedures than in the classical literature.
- Transparent reporting of prior choices, model specification, and convergence diagnostics is essential but adds to analytical complexity.
Common pitfalls
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Applications
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Frequently asked
What distinguishes fuzzy RD from sharp RD?
In a sharp design, treatment is perfectly determined by whether the running variable crosses the threshold; compliance is 100%. In a fuzzy design, the threshold causes a jump in the probability of treatment but not a jump from 0 to 1. The fuzzy estimator therefore requires a first-stage model and identifies the LATE for compliers rather than the ATE for all units.
Why use Bayesian inference instead of classical local polynomial RD?
Bayesian Fuzzy RD is most advantageous when the sample near the threshold is small. Classical asymptotic confidence intervals can be poorly calibrated in small samples, while the Bayesian posterior remains valid and fully characterises uncertainty. It also allows incorporation of prior knowledge about the outcome surface or compliance rates.
What is the LATE in this context?
The Local Average Treatment Effect is the average effect of treatment for units that comply with the threshold assignment — those who take up treatment only because they crossed the cut-off. It is local in two senses: restricted to compliers, and evaluated at the threshold rather than averaged over the full population.
How sensitive are results to prior choice?
Potentially quite sensitive when observations near the threshold are sparse, because the likelihood provides little information and the posterior is shaped largely by the prior. Standard practice is to report estimates under two or three alternative prior specifications and flag any conclusions that change substantially.
Is a manipulation check still required in the Bayesian framework?
Yes. The identification assumption — that units cannot precisely manipulate the running variable to land just above or below the threshold — is a design assumption, not a statistical one. McCrary's density test or visual inspection of the running-variable histogram should always be reported before interpreting any RD estimate, Bayesian or classical.
Sources
- 1.Hahn, J., Todd, P., & Van der Klaauw, W. (2001). Identification and Estimation of Treatment Effects with a Regression-Discontinuity Design. Review of Economic Studies, 68(1), 201-209.
- 2.Chib, S., & Jacobi, L. (2016). Bayesian fuzzy regression discontinuity analysis and returns to compulsory schooling. Journal of Applied Econometrics, 31(6), 1026-1047.
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Cite this page
ScholarGate. (2026, June 3). Bayesian Fuzzy Regression Discontinuity. ScholarGate. https://scholargate.app/causal-inference/bayesian-fuzzy-regression-discontinuity