On the geometric properties of finite mixture models
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In this paper we relate the geometry of extremal points to properties of mixtures of distributions. For a mixture model in R^J we consider as a prior the mixing density given by a uniform draw of n points from the unit (J-1)-simplex, with J ≤ n. We relate the extrema of these n points to a mixture model with m ≤ n mixture components. We first show that the extrema of the points can recover any mixture density in the convex hull of the the n points via the Choquet measure. We then show that as the number of extremal points go 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 Pólya tree. We close with an application of our model to population genomics.
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