Posterior contraction of the population polytope in finite admixture models
This work addresses theoretical convergence issues in admixture models, which are used in fields like genetics and topic modeling, but it is incremental as it builds on existing asymptotic tools.
The paper tackles the problem of understanding how the posterior distribution of the latent population structure in admixture models contracts as data increases, establishing rates of contraction with respect to Hausdorff and minimum matching Euclidean metrics on polytopes.
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.