A Unified Bayesian Nonparametric Framework for Ordinal, Survival, and Density Regression Using the Complementary Log-Log Link

TL;DR

This paper introduces a unified Bayesian nonparametric framework utilizing the complementary log-log link for ordinal, survival, and density regression, emphasizing computational efficiency and theoretical robustness.

Contribution

It develops a novel Bayesian nonparametric approach with the cloglog link, integrating ordinal, survival, and density regression models using Bayesian additive regression trees.

Findings

Efficient computational methods for ordinal regression models.

Construction of a weight-dependent Dirichlet process mixture model.

Analysis of posterior contraction rates for the models.

Abstract

In this work, we develop applications of the complementary log-log (cloglog) link to problems in Bayesian nonparametrics. Although less commonly used than the probit or logit links, we find that the cloglog link is computationally and theoretically well-suited to several commonly used Bayesian nonparametric methods. Our starting point is a Bayesian nonparametric model for ordinal regression. We first review how the cloglog link uniquely sits at the intersection of the cumulative link and continuation ratio approaches to ordinal regression. Then, we develop a convenient computational method for fitting these ordinal models using Bayesian additive regression trees. Next, we use our ordinal regression model to build a Bayesian nonparametric stick-breaking process and show that, under a proportional hazards assumption, our stick-breaking process can be used to construct a weight-dependent Dirichlet process mixture model. Again, Bayesian additive regression trees lead to convenient computations. We then extend these models to allow for Bayesian nonparametric survival analysis in both discrete and continuous time. Our models have desirable theoretical properties, and we illustrate this analyzing the posterior contraction rate of our ordinal models. Finally, we demonstrate the practical utility of our cloglog models through a series of illustrative examples.