Survival Permanental Processes for Survival Analysis with Time-Varying Covariates

Survival or time-to-event data with time-varying covariates are common in practice, and exploring the non-stationarity in covariates is essential to accurately analyzing the nonlinear dependence of time-to-event outcomes on covariates. Traditional survival analysis methods such as Cox proportional hazards model have been extended to address the time-varying covariates through a counting process formulation, although sophisticated machine learning methods that can accommodate time-varying covariates have been limited. In this paper, we propose a non-parametric Bayesian survival model to analyze the nonlinear dependence of time-to-event outcomes on time-varying covariates. We focus on a computationally feasible Cox process called permanental process, which assumes the square root of hazard function to be generated from a Gaussian process, and tailor it for survival data with time-varying covariates. We verify that the proposed model holds with the representer theorem, a beneficial property for functional analysis, which offers us a fast Bayesian estimation algorithm that scales linearly with the number of observed events without relying on Markov Chain Monte Carlo computation. We evaluate our algorithm on synthetic and real-world data, and show that it achieves comparable predictive accuracy while being tens to hundreds of times faster than state-of-the-art methods.