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.

翻译：超度量矩阵作为潜变量树模型中协方差矩阵出现，用于处理具有层次相关组分的多变量数据。作为模型中的参数空间，超度量矩阵的集合既非凸集也非光滑流形，现有文献主要局限于通过投影和松弛技术进行估计。利用超度量矩阵与有根树之间的联系，我们基于系统发育树的成熟几何结构为超度量矩阵集合赋予便捷的几何性质，超度量矩阵集合继承其优良特性（如唯一测地线和弗雷歇均值）。由此得到超度量矩阵在树空间坐标下的新表示，进而用于在贝叶斯模型中定义超度量矩阵集合上一类马尔可夫且一致的先验分布，并开发一种高效算法从后验分布中采样，该算法通过沿超度量矩阵集合内测地线进行内蕴局部移动来生成更新。模拟研究表明，所提算法能够恢复底层矩阵，后验样本以高真实拓扑频率恢复树结构，并生成具有高名义覆盖率的逐元素可信区间。我们将该算法应用于临床前癌症数据，通过构建潜在治疗树探究机制相似性，识别出与生物文献中相关通路具有高机制相似性的治疗手段。