Bayesian Multiscale Analysis of the Cox Model

TL;DR

This paper develops a comprehensive theoretical framework for Bayesian Cox models with piecewise constant priors, establishing asymptotic properties, confidence band optimality, and validating through simulations.

Contribution

It provides the first unified theory for posterior contraction, asymptotic normality, and confidence band optimality in Bayesian Cox models with histogram priors.

Findings

Posterior contraction rates are established for histogram priors.

Bernstein--von Mises theorems are proved for linear functionals of the hazard.

Bayesian credible bands for the survival function are shown to be optimal frequentist confidence bands.

Abstract

Piecewise constant priors are routinely used in the Bayesian Cox proportional hazards model for survival analysis. Despite its popularity, large sample properties of this Bayesian method are not yet well understood. This work provides a unified theory for posterior distributions in this setting, not requiring the priors to be conjugate. We first derive contraction rate results for wide classes of histogram priors on the unknown hazard function and prove asymptotic normality of linear functionals of the posterior hazard in the form of Bernstein--von Mises theorems. Second, using recently developed multiscale techniques, we derive functional limiting results for the cumulative hazard and survival function. Frequentist coverage properties of Bayesian credible sets are investigated: we prove that certain easily computable credible bands for the survival function are optimal frequentist confidence bands. We conduct simulation studies that confirm these predictions, with an excellent behavior particularly in finite samples. Our results suggest that the Bayesian approach can provide an easy solution to obtain both the coefficients estimate and the credible bands for survival function in practice.