Consider the problem of estimating the branch lengths in a symmetric 2-state substitution model with a known topology and a general, clock-like or star-shaped tree with three leaves. We show that the maximum likelihood estimates are analytically tractable and can be obtained from pairwise sequence comparisons. Furthermore, we demonstrate that this property does not generalize to larger state spaces, more complex models or larger trees. Our arguments are based on an enumeration of the free parameters of the model and the dimension of the minimal sufficient data vector. Our interest in this problem arose from discussions with our former colleague Freddy Bugge Christiansen.

中文翻译：

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对于三个序列和对称 2 状态替换模型，最大似然估计和自然成对估计方程是相同的

考虑在具有已知拓扑和具有三叶的一般时钟状或星形树的对称二状态替代模型中估计分支长度的问题。我们证明最大似然估计在分析上是易于处理的，并且可以从成对序列比较中获得。此外，我们证明该属性不能推广到更大的状态空间、更复杂的模型或更大的树。我们的论点基于模型自由参数的枚举和最小足够数据向量的维度。我们对这个问题的兴趣源于与我们的前同事 Freddy Bugge Christiansen 的讨论。