  Ultrametric matrices arise as covariance matrices in latent tree models for
multivariate data with hierarchically correlated components. As a parameter
space in a model, the set of ultrametric matrices is neither convex nor a
smooth manifold, and focus in literature has hitherto mainly been restricted to
estimation through projections and relaxation-based techniques. Leveraging the
link between an ultrametric matrix and a rooted tree, we equip the set of
ultrametric matrices with a convenient geometry based on the well-known
geometry of phylogenetic trees, whose attractive properties (e.g. unique
geodesics and Fr\'{e}chet means) the set of ultrametric matrices inherits. This
results in a novel representation of an ultrametric matrix by coordinates of
the tree space, which we then use to define a class of Markovian and consistent
prior distributions on the set of ultrametric matrices in a Bayesian model, and
develop an efficient algorithm to sample from the posterior distribution that
generates updates by making intrinsic local moves along geodesics within the
set of ultrametric matrices. In simulation studies, our proposed algorithm
restores the underlying matrices with posterior samples that recover the tree
topology with a high frequency of true topology and generate element-wise
credible intervals with a high nominal coverage rate. We use the proposed
algorithm on the pre-clinical cancer data to investigate the mechanism
similarity by constructing the underlying treatment tree and identify
treatments with high mechanism similarity also target correlated pathways in
biological literature.
