Abstract
Random measures are the key ingredient for effective nonparametric Bayesian modeling of time-to-event data. This paper focuses on priors for the hazard rate function, a popular choice being the kernel mixture with respect to a gamma random measure. Sampling schemes are usually based on approximations of the underlying random measure, both a priori and conditionally on the data. Our main goal is the quantification of approximation errors through the Wasserstein distance. Though easy to simulate, the Wasserstein distance is generally difficult to evaluate, making tractable and informative bounds essential. Here we accomplish this task on the wider class of completely random measures, yielding a measure of discrepancy between many noteworthy random measures, including the gamma, generalized gamma and beta families. By specializing these results to gamma kernel mixtures, we achieve upper and lower bounds for the Wasserstein distance between hazard rates, cumulative hazard rates and survival functions.
Highlights
One of the most attractive features of the Bayesian nonparametric approach to statistical inference is the modeling flexibility implied by priors with large support
There are several classes of priors where this property is complemented by analytical tractability, contributing to making Bayesian nonparametrics very popular in several applied areas
A prominent model for exchangeable time-to-event data is the extended gamma process for hazard rates [17], which allows for continuous observables and has been further generalized to kernel mixtures in Lo and Weng [37] and James [32]
Summary
One of the most attractive features of the Bayesian nonparametric approach to statistical inference is the modeling flexibility implied by priors with large support. We determine bounds for the Wasserstein distance between so-called completely random measures, since they act as building blocks of most popular nonparametric priors This is carried out by relying on results in Mariucci and Reiß [39] on Levy processes. We move on to using these bounds in order to quantify the divergence between hazard rate mixture models that are used to analyze time-to-event data These are applied to evaluate the approximation error in a posterior sampling scheme for the hazards, in multiplicative intensity models, that relies on an algorithm for extended gamma processes [1]. Proofs of the main results are deferred to Section 6
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
