Reliable Bayesian Causal Inference for Externally Controlled Single-Arm Trials Under Transportability Violations

Externally controlled single-arm trials are increasingly popular because they do not require randomized controls, thereby substantially reducing financial costs; however, valid causal inference in this setting relies critically on transportability between the trial and external populations, an assumption that is untestable. Frequentist semiparametrically efficient estimators, such as the doubly robust estimator, are typically analyzed under first-order asymptotics and are designed to mitigate nuisance-estimation bias under correct specification; however, this framework does not account for structural bias arising under local transportability violations, where bias and stochastic fluctuation occur at the same parametric rate. We propose a reliable Bayesian framework for estimating the average treatment effect on the treated (ATT) that combines synthetic outcome construction with residualized power-prior borrowing. Flexible machine learning is first used to construct covariate-aligned synthetic controls, after which Bayesian inference on residualized outcomes allows external borrowing to modulate uncertainty without distorting covariate-driven heterogeneity. The posterior admits a heavy-tailed distribution-unlike classical Wald procedures-and the resulting estimator lies outside the class of regular asymptotically linear estimators, thereby revealing behavior beyond first-order semiparametric asymptotics. We establish a reliability-efficiency trade-off under local transportability violations. Simulations and a case study on schizophrenia trial design demonstrate the theoretical findings and practical utility.