Bayesian X-Learner: Calibrated Posterior Inference for Heterogeneous Treatment Effects under Heavy-Tailed Outcomes
A new method delivers calibrated uncertainty and robustness to outliers simultaneously.
Eichi Uehara introduces the Bayesian X-Learner, a novel meta-learner for Conditional Average Treatment Effect (CATE) estimation that simultaneously addresses three practical requirements: heterogeneous effects τ(x), calibrated posterior uncertainty, and robustness to heavy-tailed outcome distributions. Built on the X-Learner architecture with cross-fitted doubly robust pseudo-outcomes (Kennedy, 2020), it employs a full MCMC posterior over τ(x) via a Welsch redescending pseudo-likelihood. On Hill's IHDP benchmark, the default configuration achieves a mean √ε_PEHE of 0.56 across 5 replications, matching performance of state-of-the-art methods like Causal BART and causal forest (no significant difference at α=0.05).
For heavy-tailed scenarios, the method includes a one-flag extension (contamination_severity) that selects a Huber-δ loss based on Huber's minimax-δ relation. On contaminated 'whale' data with up to 20-25% tail density, the Bayesian X-Learner attains RMSE ≈ 0.13 with tight credible intervals (single-cross-fit 30-seed coverage 83% at 20% density). A modular-Bayes pooling approach using Bayesian-bootstrap nuisance draws restores nominal 95% coverage. The paper includes 47 pages, 7 figures, 25 tables, and code is available. This work addresses a key gap: no existing widely-used tool provides all three properties (heterogeneity, calibration, heavy-tail robustness) simultaneously.
- Achieves mean √ε_PEHE = 0.56 on IHDP benchmark, competitive with Causal BART and causal forest
- On contaminated data with 20% tail density, RMSE ≈ 0.13 with 83% coverage; modular-Bayes pooling restores 95% coverage
- Combines cross-fitted doubly robust pseudo-outcomes with MCMC posterior via Welsch pseudo-likelihood and optional Huber-δ loss
Why It Matters
Enables reliable causal inference in real-world data with outliers, critical for medicine, economics, and policy.