Posterior Consistency for a Non-parametric Survival Model under a Gaussian Process Prior

TL;DR

This paper proves the almost sure posterior consistency of a non-parametric survival model using Gaussian process priors on hazard functions, applicable to multivariate covariates and extending Schwartz's theorem.

Contribution

It establishes posterior consistency for a Gaussian process-based survival model with a new metric and extends Schwartz's theorem to non-i.i.d. data in survival analysis.

Findings

Proves posterior consistency under the new metric.

Extends Schwartz's theorem for non-i.i.d. data.

Provides Gaussian process results on b3+ for potential broader use.

Abstract

In this paper, we prove almost surely consistency of a Survival Analysis model, which puts a Gaussian process, mapped to the unit interval, as a prior on the so-called hazard function. We assume our data is given by survival lifetimes $T$ belonging to $\mathbb{R}^{+}$, and covariates on $[0,1]^d$, where $d$ is an arbitrary dimension. We define an appropriate metric for survival functions and prove posterior consistency with respect to this metric. Our proof is based on an extension of the theorem of Schwartz (1965), which gives general conditions for proving almost surely consistency in the setting of non i.i.d random variables. Due to the nature of our data, several results for Gaussian processes on $\mathbb{R}^+$ are proved which may be of independent interest.