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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.

Nonparametric latent structure models provide flexible inference on distinct, yet related, groups of observations. Each component of a vector of d ≥ 2 random measures models the distribution of a group of exchangeable observations, while their dependence structure regulates the borrowing of information across different groups. Recent work has quantified the dependence between random measures in terms of Wasserstein distance from the maximally dependent scenario when d = 2. By solving an intriguing max-min problem we are now able to define a Wasserstein index of dependence I W with the following properties: (i) it simultaneously quantifies the dependence of d ≥ 2 random measures; (ii) it takes values in [0,1]; (iii) it attains the extreme values {0, 1} under independence and complete dependence, respectively; (iv) since it is defined in terms of the underlying Lévy measures, it is possible to evaluate it numerically in many Bayesian nonparametric models for partially exchangeable data.