We study the problem of maximum likelihood estimation given one data sample ($n=1$) over Brownian Motion Tree Models (BMTMs), a class of Gaussian models on trees. BMTMs are often used as a null model in phylogenetics, where the one-sample regime is common. Specifically, we show that, almost surely, the one-sample BMTM maximum likelihood estimator (MLE) exists, is unique, and corresponds to a fully observed tree. Moreover, we provide a polynomial time algorithm for its exact computation. We also consider the MLE over all possible BMTM tree structures in the one-sample case and show that it exists almost surely, that it coincides with the MLE over diagonally dominant M-matrices, and that it admits a unique closed-form solution that corresponds to a path graph. Finally, we explore statistical properties of the one-sample BMTM MLE through numerical experiments.

翻译：我们研究对布朗动树模型(BMTMs)进行一个数据抽样(n=1美元)的最大可能性估算问题,BMTMs是树上高斯模型的一类。BMTMs经常在植物性模型中作为无效模型使用,而一模一样的制度是常见的。具体地说,我们表明,几乎可以肯定,一模一样的BMTM最大可能性估测仪(MMLE)的存在是独一无二的,与完全观察的树相对应。此外,我们为精确计算提供了一种多元时间算法。我们还在一模版案例中对所有可能的BMMTM树结构进行考虑,并表明它几乎肯定地存在,它与MLE超过对等占优势M-矩阵的M-矩阵相一致,而且它承认一个与路径图相对应的独特封闭式的解决方案。最后,我们通过数字实验来探索单模的BMMMMLE的统计属性。