Infinitely divisible priors for multivariate survival functions

This article introduces a novel framework for nonparametric priors on real-valued random vectors. The framework can be viewed as a multivariate generalization of neutral-to-the right priors, which also encompasses continuous priors. It is based on randomizing the exponent measure of a minimum-infinitely divisible random vector by an infinitely divisible random measure and naturally incorporates partially exchangeable data as well as exchangeable random vectors. We show how to construct hierarchical priors from simple building blocks and embed many models from Bayesian nonparametric survival analysis into our framework. The prior can concentrate on discrete or continuous distributions and other properties such as dependence, moments and moments of mean functionals are characterized. The posterior predictive distribution is derived in a general framework and is refined under some regularity conditions. In addition, a theoretical framework for the simulation from the posterior predictive distribution is provided. As a byproduct, the concept of subordination of homogeneous completely random measures by homogeneous completely random measures is extended to subordination by infinitely divisible random measures. This technique allows to create vectors of dependent infinitely divisible random measures with tractable Laplace transform and is expected to have applications beyond the scope of this paper.