Bayesian Causal Stress Testing: Posterior Fragility of Treatment-Effect Conclusions

Bayesian causal inference quantifies uncertainty in treatment effects under a specified causal model. This uncertainty is often interpreted as evidence that the resulting causal conclusion is robust. We argue these are distinct targets: a posterior over a treatment effect may be tightly concentrated above zero while the corresponding causal claim is fragile to small violations of ignorability, overlap, or target-population assumptions. We introduce Bayesian causal stress testing (BCS), a framework that assigns a posterior fragility score $F_\alpha$ with a threshold $\alpha\in(0,1)$ to a causal conclusion by measuring the smallest calibrated stress under which the posterior probability of the claim falls below $\alpha$. We give a stress-map formulation, decision and composition results for claim-level stress, a calibration theorem showing that $F_{0.5}$ asymptotically recovers the partial-correlation form of the Cinelli-Hazlett robustness value in the linear-Gaussian flat-prior limit, and a proposition formalizing why posterior precision and posterior fragility can decouple. We evaluate BCS on IHDP, TWINS, and ACIC-cov+Hill, a semi-synthetic benchmark built from the ACIC 2016 covariate matrix, under BART, BCF-style, Bayesian linear, and Bayesian-bootstrap AIPW posteriors. On the linear Bayesian posterior, $F_{0.5}^2$ matches the Cinelli-Hazlett $RV_{0.5}^{\mathrm{CH}}$ to RMS $0.001$ on both IHDP and ACIC-cov+Hill across all $20$ configurations; the BART and BCF-style posteriors show larger deviations in small-effect configurations. Across estimators, posterior $95$% interval widths vary by up to $50$% but $F_{0.5}$ remains within $0.05$, illustrating that posterior precision is not a robustness certificate. A controlled-DGP calibration experiment with a hidden confounder of known partial correlation $\gamma_{\mathrm{true}}$ confirms that $F_{0.5}$ tracks the required flip-stress to mean absolute error $0.03$.