Molecular phylogenies have been increasingly recognized as an important source of information on species diversification. For many models of macroevolution, analytical likelihood formulas have been derived to infer macroevolutionary parameters from phylogenies. A few years ago, a general framework to numerically compute such likelihood formulas was proposed, which accommodates models that allow speciation and/or extinction rates to depend on diversity. This framework calculates the likelihood as the probability of the diversification process being consistent with the phylogeny from the root to the tips. However, while some readers found the framework presented in Etienne et al. (Proc R Soc Lond B Biol Sci 279(1732):1300–1309, 2012) convincing, others still questioned it (personal communication), despite numerical evidence that for special cases the framework yields the same (i.e., within double precision) numerical value for the likelihood as analytical formulas do that were independently derived for these special cases. Here we prove analytically that the likelihoods calculated in the new framework are correct for all special cases with known analytical likelihood formula. Our results thus add substantial mathematical support for the overall coherence of the general framework.

中文翻译：

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对计算多元化模型可能性的新方法的额外分析支持

分子系统发育越来越被认为是物种多样化的重要信息来源。对于许多宏观进化模型，已经推导出分析似然公式以从系统发育推断宏观进化参数。几年前，提出了一个数值计算此类似然公式的通用框架，该框架适用于允许物种形成和/或灭绝率取决于多样性的模型。该框架将可能性计算为多样化过程与从根到尖端的系统发育一致的概率。然而，虽然一些读者发现了 Etienne 等人提出的框架。（Proc R Soc Lond B Biol Sci 279(1732):1300–1309, 2012）令人信服，其他人仍然质疑它（个人交流），尽管有数字证据表明，对于特殊情况，该框架产生的似然数值与针对这些特殊情况独立推导出的解析公式相同（即在双精度范围内）。在这里，我们通过分析证明新框架中计算的似然对于所有具有已知分析似然公式的特殊情况都是正确的。因此，我们的结果为总体框架的整体一致性增加了大量的数学支持。