Finite mixture models: a bridge with stochastic geometry and Choquet theory

In Bayesian density estimation, a question of interest is how the number of components in a finite mixture model grows with the number of observations. We provide a novel perspective on this question by using results from stochastic geometry to find that the growth rate of the expected number of components of a finite mixture model whose components belong to the unit simplex $\Delta^{J-1}$ of the Euclidean space $\mathbb{R}^J$ is $(\log n)^{J-1}$. We also provide a central limit theorem for the number of components. In addition, we relate our model to a classical non-parametric density estimator based on a P\'olya tree. Combining this latter with techniques from Choquet theory, we are able to retrieve mixture weights. We also give the rate of convergence of the P\'olya tree posterior to the Dirac measure on the weights. The analyses in this paper apply to the well developed and popular latent Dirichlet allocation model.