On the expected number of components in a finite admixture model
Michele Caprio, Sayan Mukherjee · arXiv (Cornell University) · 2020
In Bayesian non-parametric density estimation a question of interest is how the number of components in the model grows with the number of observations. We state the growth rate of the number of components for a finite admixture model both in expectation and in distribution. The tools we use in our analysis combine the concept of a Choquet measure with classic results in stochastic geometry on the number of extrema of random polytopes. We show that if our admixture weights are probability vectors from a unit $(J-1)$-simplex then the number of admixture components grows as $(\log n)^{J-1}$; in the standard mixture case we recover the $\log n$ rate. We also state a central limit theorem for the number of mixture components. In addition, we state the convergence of the sequence of the empirical measures generated by our model to the Choquet measure. Lastly, we relate our model to a classical non-parametric density estimator based on a Polya tree.