Posterior contraction of the population polytope in finite admixture models

We study the posterior contraction behavior of the latent population structure that arises in admixture models as the amount of data increases. An admixture model — alternatively known as a topic model — specifies k populations, each of which is characterized by a ∆-valued vector of frequencies for generating a set of discrete values in {0, 1, . . . , d}. The population polytope is defined as the convex hull of the k frequency vectors. Under the admixture specification, each of m individuals generates an i.i.d. frequency vector according to a probability distribution defined on the (unknown) population polytope G0, and then generates n data points according to the sampled frequency vector. Rates of posterior contraction are established with respect to Hausdorff metric and a minimum matching Euclidean metric defined on population polytopes, as the amount of data m × n tends to infinity. Minimax lower bounds are also established. Tools developed include posterior asymptotics of hierarchical models with m× n data, and arguments from convex geometry.