B-CALM: Bias-Limited Bayesian Borrowing for RCT-Anchored Treatment Effects under Covariate Mismatch

Randomized controlled trials (RCTs) identify trial-anchored treatment effects but are often too small for reliable heterogeneity estimation; observational studies (OS) are larger but confounded and only partially overlap the trial in measured covariates. We propose Bayesian Calibrated ALignment under covariate Mismatch (B-CALM), a Bayesian borrowing framework for RCT-anchored conditional average treatment effect (CATE) estimation. B-CALM maps source-specific covariates into a shared latent state, jointly models trial and observational outcome surfaces, and introduces baseline-bias and comparative-bias functions that absorb how the OS departs from the trial estimand. The comparative-bias prior becomes an explicit sensitivity knob: we prove a function-valued bias-limited information bound showing that observational contrast information about the trial treatment-effect surface is capped by the prior precision of this bias function, with a scalar corollary in which the effective sample size (ESS) saturates as OS sample size grows. A PAC-Bayes-style risk decomposition separates RCT empirical risk, latent alignment, and residual calibration of the debiased OS surface. Across synthetic, semi-synthetic, and pediatric-obesity external-control studies, B-CALM delivers calibrated credible intervals and low negative transfer while pooled and forest baselines can become overconfident under comparative bias.