Magnitude, concentration, and metric complexity in phylogenetics and information theory

The results in this thesis contribute to two areas: phylogenetic inference and information theory. In the first part, we investigate some probabilistic models and problems that arise when one wants to estimate evolutionary parameters from large biological datasets. The classical setting for inference assumes that the number of independent sites, k, is much larger than the number of leaves, n, on a phylogenetic tree. In Chapter 2, we add to the list of examples of how estimation can go wrong when k is much smaller than n. We show that on the balanced claw tree under the binary non-symmetric model, the maximum likelihood estimator of evolutionary parameters is inconsistent when k = 1. ☐ To prove results about consistency in the classical case, concentration inequalities for independent random variables are used. However, when k