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

In the context of a finite mixture model whose components and weights are unknown, if the number of components is a function of the amount of data collected, we give the growth rate of its expected value. We also show that by placing a Dirichlet process prior on the densities supported on the unit simplex, we are able to retrieve the Dirac measure at the Choquet measure supported on the components. In turn, this gives us the weights. Finally, we propose a novel algorithm that identifies the model capturing the complexity of the data using only the strictly necessary number of components.