Abstract
Conditional Average Treatment Effect (CATE) estimation in practice demands three properties simultaneously: heterogeneous effects \tau(x), calibrated uncertainty over them, and robustness to the heavy tails that contaminate real outcome data. Meta-learners (K\"unzel et al., 2019) give (i); causal forests and BART give (i)-(ii) with Gaussian-tail assumptions; no widely used tool gives all three. We present Bayesian X-Learner, an X-Learner built on cross-fitted doubly robust pseudo-outcomes (Kennedy, 2020) with a full MCMC posterior over \tau(x) via a Welsch redescending pseudo-likelihood. On Hill's IHDP benchmark the default configuration attains mean \sqrt{\varepsilon_{\mathrm{PEHE}}} = 0.56 on 5 replications (lowest mean; differences from S-/T-/X-learners, full-config Causal BART, and a causal forest baseline are not significant at \alpha=0.05, and rank ordering is unstable at 10 replications -- IHDP comparisons are competitive rather than dominant). On contaminated "whale" DGPs with up to 20-25% tail density, a one-flag extension (contamination_severity) that selects a Huber-\delta nuisance loss per Huber's minimax-\delta relation recovers RMSE \approx 0.13 with tight credible intervals (single-cross-fit 30-seed coverage 83% [Wilson 66%, 93%] at 20% density; modular-Bayes pooling with Bayesian-bootstrap nuisance draws restores nominal 95% coverage).
