Multi-resolution two-sample comparison through the divide-merge Markov tree

We introduce a probabilistic framework for two-sample comparison based on a nonparametric process taking the form of a Markov model that transitions between a "divide" and a "merge" state on a multi-resolution partition tree of the sample space. Multi-scale two-sample comparison is achieved through inferring the underlying state of the process along the partition tree. The Markov design allows the process to incorporate spatial clustering of differential structures, which is commonly observed in two-sample problems but ignored by existing methods. Inference is carried out under the Bayesian paradigm through recursive propagation algorithms. We demonstrate the work of our method through simulated data and a real flow cytometry data set, and show that it substantially outperforms other state-of-the-art two-sample tests in several settings.