On the expected number of components in a finite mixture model.

In this paper we describe the growth rate of the expected number of components in a finite mixture model. We do so by relating the geometry of extremal points to finite mixtures of distributions. In particular, a finite mixture model in $\mathds{R}^J$ with $m$ components can be represented by an element of the convex hull of $n$ points drawn uniformly from the unit $(J-1)$-simplex, $J \leq m \leq n$. We also inspect the theoretical properties of this model: we first show that the extrema of the convex hull can recover any mixture density in the convex hull via the Choquet measure. We then show that as the number of extremal points goes to infinity, the convex hull converges to a smooth convex body. We also state a Central Limit Theorem for the number of extremal points. In addition, we state the convergence of the sequence of the empirical measures generated by our model to the Choquet measure. We relate our model to a classical non-parametric one based on a P\'olya tree. We close by applying our model to a classic admixture model in population and statistical genetics.