Bayesian mixtures models with repulsive and attractive atoms

The study of almost surely discrete random probability measures is an active line of research in Bayesian non-parametrics. The idea of assuming interaction across the atoms of the random probability measure has recently spurred significant interest in the context of Bayesian mixture models. This allows the definition of priors that encourage well-separated and interpretable clusters. In this work, we provide a unified framework for the construction and the Bayesian analysis of random probability measures with interacting atoms, encompassing both repulsive and attractive behaviours. Specifically, we derive closed-form expressions for the posterior distribution, the marginal and predictive distributions, previously unavailable except for the case of measures with i.i.d. atoms. We show how these quantities are fundamental for both prior elicitation and developing new posterior simulation algorithms for hierarchical mixture models. Our results are obtained without any assumption on the finite point process governing the atoms of the random measure. Their proofs rely on analytical tools borrowed from Palm calculus theory, which might be of independent interest. We specialize our treatment to the classes of Poisson, Gibbs, and determinantal point processes, as well as in the case of shot-noise Cox processes. Finally, we illustrate different modelling strategies on simulated and real datasets.