Bayesian mixture models with repulsive and attractive atoms

The study of almost surely discrete random probability measures is an active line of research in Bayesian nonparametrics. 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.