Bayesian Hierarchical Piecewise-Exponential Cause-Specific Hazards for Posterior Predictive Validation of Competing-Risks Models

Competing-risks prediction requires validation methods that assess predictive support for both event time and event type. This is difficult for cause-specific hazard models because out-of-sample log-scores require positive, well-defined hazards over the validation horizon, and rare event causes can lead to unstable cause-specific estimates. We develop the Bayesian hierarchical piecewise-exponential cause-specific hazards model ( $\text{BH-PE-CSH}$ ), a competing-risks model designed for posterior predictive validation using cross-validated log-scores. The model specifies positive piecewise-exponential baseline hazards for each cause and places hierarchical priors on cause-specific regression coefficients, allowing covariate effects to borrow information across causes while retaining cause-specific interpretation. Predictive performance is evaluated using cross-validated posterior predictive log-scores under the cause-specific hazards representation. Simulation studies show that the Bayesian piecewise-exponential framework improves out-of-sample predictive support and model-selection performance across settings with varying signal strength, sample size, censoring, cause imbalance, and cross-cause heterogeneity. Hierarchical pooling provides the largest benefit when cross-cause heterogeneity is low to moderate, event causes are imbalanced, or information for rare causes is limited. When true cause-specific effects are highly heterogeneous, the advantage of partial pooling diminishes, reflecting the expected bias-variance trade-off. A Surveillance, Epidemiology, and End Results (SEER) colorectal cancer application further demonstrates the practical utility of the method: the treatment-extended specification evaluated with $\text{BH-PE-CSH}$ achieves the best predictive performance, and hierarchical pooling narrows posterior credible intervals for selected rare-cause coefficients. These findings support $\text{BH-PE-CSH}$ as a stable Bayesian posterior predictive framework for imbalanced multi-cause competing-risks prediction.