# Bayesian Elastic Net Cox Models for Time-to-Event Prediction: Application to a Breast Cancer Cohort

## TL;DR

This paper introduces a Bayesian version of the elastic net Cox model for predicting survival outcomes in breast cancer patients, offering better uncertainty estimates and improved performance over existing methods.

## Contribution

The novel Bayesian elastic net Cox model introduces a hierarchical shrinkage prior and Hamiltonian Monte Carlo inference for survival analysis with high-dimensional data.

## Key findings

- BEN–Cox achieves better prediction error and calibration than ridge, lasso, and elastic net Cox baselines on a breast cancer cohort.
- The model identifies a compact and biologically plausible gene panel with interpretable sparse signatures.
- Posterior summaries provide credible intervals for hazard ratios and support theoretical stability of risk scores.

## Abstract

High-dimensional survival analyses require calibrated risk and measurable uncertainty, but standard elastic net Cox models provide only point estimates. We develop a Bayesian elastic net Cox (BEN–Cox) model for 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; using HMC with the Cox partial likelihood, we obtain full posterior distributions 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=1903; p=440 gene-level features after preprocessing, derived from an Illumina HT-12 array with ≈24,000 probes at the raw feature level), BEN–Cox achieves slightly lower prediction error, higher discrimination, and better global calibration than a tuned ridge Cox, lasso Cox, and elastic net Cox baselines on a held-out test set. Posterior summaries provide credible intervals for hazard ratios and identify a compact gene panel that remains biologically plausible. BEN–Cox provides an uncertainty-aware alternative to tuned penalized Cox models with theoretical support, offering modest improvements in calibration and providing an interpretable sparse signature in highly-correlated survival data.