Title:Repulsion, Chaos and Equilibrium in Mixture Models <br>
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Abstract: 
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 locations of the mixture. Widely used are Bayesian nonparametric models based on mixtures with infinite or random number of components. Despite their popularity, the flexibility of these models often does not translate into interpretability of the clusters. To overcome this issue, repulsive mixture models have been recently proposed. The basic idea is to include a repulsive term in the distribution of the atoms of the mixture, favouring mixture locations far apart. This approach allows one to produce well-separated clusters, aiding the interpretation of the results. However, these models are usually not easy to handle due to unknown normalising constants. Conceptually, we take inspiration from equilibrium statistical mechanics, where the molecular chaos hypothesis implies that nearby particles will be spread out over time. Technically, we exploit the connection between random matrix theory and statistical mechanics to obtain tractable prior distributions. Leveraging results from statistical mechanics, we propose a novel class of repulsive prior distributions based on Gibbs measures. Specifically, we use Gibbs measures associated with joint distributions of eigenvalues of random matrices, naturally possessing a repulsive property. The proposed repulsive priors are suitable for modelling both locations and weights. The proposed framework greatly simplifies computations thanks to the availability of the normalising constant in closed form. We investigate theoretical properties and clustering performance of the proposed distributions.
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Joint work with: Maria De Iorio, Tim Wertz, Alexander Mozdzen, Gregor Kastner