Distribution Recovery with Domain Constraints: Applications to Medical Time Series

Recovering the occurrence and severity of physiological abnormalities from intermittent clinical measurements requires a distribution over possible trajectories, yet sparse observations may not uniquely determine this distribution. SFBD-OMNI guides recovery with clean reference sample anchors, which are generally unavailable in retrospective medical records. In this work, we introduce Projected SFBD, which directly incorporates domain knowledge as constraints on recovered trajectories through a projection-based update. We provide theoretical convergence guarantees and show how physiological priors narrow the space of admissible distributions and can improve identifiability without clean reference samples. On irregular MIMIC-III charts, evaluation against bedside-monitor references shows better trajectory-distribution scores and more accurate recovery of hypotension occurrence, severity, recurrence, and within-stay variation in vital signs than strong baselines. Ablations on MIMIC-III and the Pulse simulator show that projection better preserves clinically relevant variation and yields less biased clinical summaries than constructed clean anchors.