Posterior contraction of the population polytope in finite admixture models

TL;DR

This paper investigates how the posterior distribution of the population structure in admixture models concentrates around the true structure as data increases, using geometric and convex analysis tools.

Contribution

It introduces a geometric framework for analyzing posterior contraction in admixture models and establishes convergence rates with respect to specific polytope metrics.

Findings

Posterior contraction rates are derived for the population polytope.

The geometric approach links convex geometry with Bayesian asymptotics.

Results apply to models with latent population structures.

Abstract

We study the posterior contraction behavior of the latent population structure that arises in admixture models as the amount of data increases. We adopt the geometric view of admixture models - alternatively known as topic models - as a data generating mechanism for points randomly sampled from the interior of a (convex) population polytope, whose extreme points correspond to the population structure variables of interest. Rates of posterior contraction are established with respect to Hausdorff metric and a minimum matching Euclidean metric defined on polytopes. Tools developed include posterior asymptotics of hierarchical models and arguments from convex geometry.