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 behaviors. Specifically we derive closed-form expressions for the posterior distribution, the marginal and predictive distributions, which were not previously available except for the case of measures with i.i.d. atoms. We show how these quantities are fundamental both for prior elicitation and to develop new posterior simulation algorithms for hierarchical mixture models. Our results are obtained without any assumption on the finite point process that governs the atoms of the random measure. Their proofs rely on new analytical tools borrowed from the theory of Palm calculus and that might be of independent interest. We specialize our treatment to the classes of Poisson, Gibbs, and Determinantal point processes, as well as to the case of shot-noise Cox processes.

翻译：几乎必然离散的随机概率测度是贝叶斯非参数领域的一个活跃研究方向。近期，在贝叶斯混合模型背景下，假设随机概率测度原子之间存在交互作用的思路引发了显著关注。这种思路允许定义能够促进聚类结果良好分离且可解释的先验分布。本文构建了一个统一框架，用于构造和进行带交互原子（涵盖排斥与吸引行为）的随机概率测度的贝叶斯分析。具体而言，我们推导了后验分布、边缘分布和预测分布的闭式表达式——这些结果此前仅对具有独立同分布原子的测度可用。我们展示了这些量在先验指定以及为层次混合模型开发新型后验模拟算法中的基础性作用。我们的结果不依赖于对控制随机测度原子的有限点过程做出任何假设。其证明过程借助了源于Palm计算理论的新型分析工具，这些工具可能具有独立的研究价值。我们将处理方法专门应用于泊松点过程、吉布斯点过程、行列式点过程以及散粒噪声Cox过程等类别。