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, 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 analytical tools borrowed from the Palm calculus theory, which might be of independent interest. We specialise 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 the performance of different modelling strategies on simulated and real datasets.

翻译：几乎必然离散的随机概率测度研究是贝叶斯非参数领域的一个活跃研究方向。近年来，在贝叶斯混合模型背景下，假设随机概率测度原子间存在相互作用的思想激发了显著的研究兴趣。这种思路能够定义出鼓励形成分离良好且可解释聚类簇的先验分布。本研究提出了一个统一框架，用于构建具有相互作用原子的随机概率测度并进行贝叶斯分析，该框架同时涵盖排斥与吸引两种行为模式。具体而言，我们推导出了后验分布、边缘分布与预测分布的闭式表达式——这些表达式在原子独立同分布的情形之外均未得到过解析表示。我们论证了这些量值对于先验设定以及开发分层混合模型的新型后验模拟算法具有基础性意义。所得结果无需对支配随机测度原子的有限点过程作任何假设，其证明过程借鉴了源自帕姆微积分理论的解析工具，这些工具本身可能具有独立的研究价值。我们将该理论框架具体应用于泊松点过程、吉布斯点过程、行列式点过程以及散粒噪声考克斯过程等类别。最后，我们通过模拟数据集与真实数据集展示了不同建模策略的性能表现。