Bayesian Elastic‑Net Cox Models for Time‑to‑Event Prediction: Application with Breast‑Cancer Cohort

High-dimensional survival analyses require calibrated risk and honest uncertainty, but standard elastic-net Cox models yield only point estimates. We develop a fully Bayesian elastic-net Cox (BEN–Cox) modelfor high-dimensional proportional hazards regression that places a hierarchical global–local shrinkage prior on coefficients and performs full Bayesian inference via Hamiltonian Monte Carlo. We represent the elastic–net penalty as a global–local Gaussian scale mixture with hyperpriors that learn the ℓ1/ℓ2 trade-off, enabling adaptive sparsity that preserves correlated gene groups and, using HMC on the Cox partial likelihood, yields full posteriors for hazard ratios and patient-level survival curves. Methodologically, we formalize a Bayesian analogue of the elastic-net grouping effect at the posterior mode and establish posterior contraction under sparsity for the Cox partial likelihood, supporting the stability of the resulting risk scores. On the METABRIC breast-cancer cohort (n = 1 , 903; 440 gene-level features from an Illumina array with ≈ 24,000 gene-level features (probes)), BEN–Cox achieves slightly lower prediction error, higher discrimination, and better global calibration than a tuned ridge Cox baseline on a held-out test set. Posterior summaries provide credible intervals for hazard ratios, identify a compact gene panel that remains biologically plausible. BEN–Cox provides a theory-backed, uncertainty-aware alternative to tuned penalised Cox models, improving calibration and yielding an interpretable sparse signature in correlated, high-dimensional survival data