Belief-Augmented Joint Models for Causal Inference of Time-Varying Treatment Effects from Sparse Observations

Identifying and estimating time-varying treatment effects from observational healthcare data is challenging when clinicians act on beliefs about patient trajectories that are only partially observed. Calibration for observed covariates alone may leave residual confounding induced by these beliefs. We propose the Belief-Augmented Joint Model (BAM), which restructs natural recovery from treatment effects by incorporating a belief-dependent model of clinical decisions. BAM represents clinicians’ beliefs through a closed-form posterior over latent trajectory coefficients, derived from sparse, noisy observations under a linear-Gaussian trajectory model. We prove that under belief-driven confounding—the latent trajectory influences the decision only through this belief—the time-varying treatment effect is identified without estimating or parameterizing the underlying decision rule. In synthetic benchmarks, BAM outperforms eight standard models; its structural decision channels transfer to a rehabilitation cohort and MIMIC-III. These results establish a framework for estimating longitudinal treatment effects that explicitly accounts for the beliefs linking incomplete patient observations to clinical decisions, empowering clinicians to take a central role in high-risk, long-cycle rehabilitation information design.