Mixture models are commonly used in applications with heterogeneity and overdispersion in the population, as they allow the identification of subpopulations. In the Bayesian framework, this entails the specification of suitable prior distributions for the weights and location parameters of the mixture. Widely used are Bayesian semi-parametric models based on mixtures with infinite or random number of components, such as Dirichlet process mixtures or mixtures with random number of components. Key in this context is the choice of the kernel for cluster identification. Despite their popularity, the flexibility of these models and prior distributions often does not translate into interpretability of the identified clusters. To overcome this issue, clustering methods based on repulsive mixtures have been recently proposed. The basic idea is to include a repulsive term in the prior distribution of the atoms of the mixture, which favours mixture locations far apart. This approach is increasingly popular and allows one to produce well-separated clusters, thus facilitating the interpretation of the results. However, the resulting models are usually not easy to handle due to the introduction of unknown normalising constants. Exploiting results from statistical mechanics, we propose in this work a novel class of repulsive prior distributions based on Gibbs measures. Specifically, we use Gibbs measures associated to joint distributions of eigenvalues of random matrices, which naturally possess a repulsive property. The proposed framework greatly simplifies the computations needed for the use of repulsive mixtures due to the availability of the normalising constant in closed form. We investigate theoretical properties of such class of prior distributions, and illustrate the novel class of priors and their properties, as well as their clustering performance, on benchmark datasets.

翻译：混合模型常用于存在异质性和过度分散性的应用中，因为它能够识别子群体。在贝叶斯框架下，这需要为混合模型的权重和位置参数设定合适的先验分布。广泛使用的贝叶斯半参数模型基于具有无限或随机成分数量的混合模型，例如狄利克雷过程混合模型或具有随机成分数量的混合模型。在此背景下，核函数的选择对聚类识别至关重要。尽管这些模型及其先验分布具有灵活性，但往往难以转化为对识别出的聚类的可解释性。为克服这一问题，近年来提出了基于排斥混合的聚类方法。其基本思想是在混合模型原子先验分布中加入排斥项，以促使混合位置彼此远离。这种方法日益流行，能够产生分离良好的聚类，从而便于结果解释。然而，由于引入未知的归一化常数，此类模型通常难以处理。借助统计力学的结果，本文提出了一类基于吉布斯测量的新型排斥先验分布。具体而言，我们利用与随机矩阵特征值联合分布相关的吉布斯测量，这类分布天然具有排斥性质。由于归一化常数具有闭式解，所提出的框架极大简化了使用排斥混合所需的计算。我们研究了这类先验分布的理论性质，并在基准数据集上展示了新先验类及其性质以及聚类性能。