DBPS: Doob-Bridge Posterior Sampler with Balanced Endpoint-Population Control for Unpaired Neurodegenerative Pathology Transport

Modeling neurodegenerative pathology from observational cohorts is challenging because subjects are typically observed as unpaired endpoint snapshots rather than aligned longitudinal trajectories. Existing approaches to pathology transport are either endpoint-conditioned or population-level, yet neither extreme alone yields the best terminal-distribution fit. We introduce the Doob-Bridge Posterior Sampler (DBPS), a controlled-SDE posterior sampler with balanced endpoint-population control that blends individual endpoint guidance with population-level guidance. This ρ-weighted controller, obtained as the Cole–Hopf log-gradient of a geometric interpolation between endpoint and population desirability functions, is the exact optimal feedback policy of a linearly-solvable stochastic optimal control problem with closed-form terminal and running costs. Endpoint-conditioned bridge sampling and population Doob sampling are the ρ = 0 and ρ = 1 boundary cases; ρ ∈ (0, 1) realizes the genuine balanced regime. We evaluate DBPS on tau PET pathology transport in Alzheimer’s disease across ADNI and NACC cohorts (84 brain regions). Compared to representative baselines from all major unpaired-transport families, DBPS achieves state-of-the-art performance across multi-cohort settings (union training: SWD 0.760 vs. 0.821; held-out ADNI: 0.822 vs. 0.859), and remains competitive in single-cohort training (ADNI: 0.868 vs. 0.876). Beyond distribution matching, the learned controls recover canonical Braak-stage pathology organization without anatomical supervision, suggesting that balanced SOC objectives capture biologically meaningful structure in latent neurodegenerative progression.