  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.
