Finite Admixture Models: a Bridge with Stochastic Geometry and Choquet Theory

Abstract

Given a finite admixture model whose components and weights are unknown, let the number of identifiable components be a function of the amount of data sampled from a known distribution on the unit simplex. We use techniques from stochastic convex geometry to find the growth rate of its expected value. In addition, when the components are known but the weights are not, we provide an application of the classic Glivenko-Cantelli's theorem that allows us to retrieve the Choquet measure supported on the identifiable admixture components. In turn, this gives us the identifiable admixture weights. Finally, we propose a novel algorithm that estimates the model capturing the complexity of the data using only the strictly necessary number of components.