Learning and Uncertainty in Generalized Bayesian Discrete-Time Survival Distillation

External survival models can improve prediction in target cohorts with few events, but the inferential consequences of borrowing from a fixed teacher remain insufficiently characterized. Existing discrete-time survival distillation methods primarily focus on point estimation; how teacher guidance affects survival-probability uncertainty requires further analysis. We develop a generalized Bayesian framework for discrete-time survival distillation that transfers interval-level conditional event probabilities from a fixed external teacher to a flexible neural student through a prediction-level Kullback–Leibler divergence, allowing teacher and student models to differ in architecture and parameterization. The resulting update combines the target survival likelihood, teacher guidance, and structural prior regularization to define a posterior over individualized survival predictions. Our local analysis quantifies how borrowing shifts fitted survival probabilities and distinguishes curvature-based posterior variance from sampling variance, which also depends on patient-level score covariance. We derive a generalized subject-deletion predictive criterion for selecting the borrowing weight. Experiments under compatible and imperfect teacher guidance show that borrowing can improve survival prediction and reduce sampling variability, whereas excessive reliance on an imperfect teacher can worsen prediction and introduce substantial bias. Real-data analyses show improved agreement with a pooled-data reference in breast cancer survival and predictive gains from borrowing across kidney-transplant populations. The framework supports flexible survival borrowing while clarifying its inferential benefits and limitations.