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研究是贝叶斯非参数领域中的一个活跃研究方向。近年来，在贝叶斯混合模型的背景下，假设随机概率测度中原子之间存在交互作用这一思路引起了广泛关注。这有助于定义能够促进良好分离且可解释聚类的先验分布。本文为构建和分析具有交互原子（涵盖排斥与吸引行为）的随机概率测度提供了统一框架。具体而言，我们推导了后验分布、边缘分布及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后验模拟算法中的基础性作用。我们的结论不依赖于对控制随机测度原子的有限点过程的任何假设。其证明借助了源自Palm演算理论的新分析工具，这些工具可能具有独立的研究价值。我们将分析具体应用于泊松点过程、吉布斯点过程和行列式点过程，以及散粒噪声Cox过程的情形。