Testing to distinguish measures on metric spaces

We study the problem of distinguishing between two distributions on a metric space; i.e., given metric measure spaces (X, d, μ1) and (X, d, μ2), we are interested in the problem of determining from finite data whether or not μ1 is μ2. The key is to use pairwise distances between observations and, employing a reconstruction theorem of Gromov, we can perform such a test using a two sample Kolmogorov–Smirnov test. A real analysis using phylogenetic trees and flu data is presented.