Transport distances on random vectors of measures: recent advances in Bayesian nonparametrics

Random vectors of measures are at the core of many recent developments in Bayesian nonparametrics. For a deep understanding of these infinite-dimensional discrete random structures and their impact on the inferential and theoretical properties of the induced models, we consider a class of transport distances based on the Wasserstein distance. The geometrical definition makes it ideal for measuring similarity between distributions with possibly different supports. Moreover, when applied to random vectors of measures with independent increments (completely random vectors), the interesting theoretical properties are coupled with analytical tractability. This leads to a new measure of dependence for completely random vectors and the quantification of the impact of hyperparameters in notable models for exchangeable time-to-event data.