The coalescent is a foundational model of latent genealogical trees under neutral evolution, but suffers from intractable sampling probabilities. Methods for approximating these sampling probabilities either introduce bias or fail to scale to large sample sizes. We show that a class of cost functionals of the coalescent with recurrent mutation and a finite number of alleles converge to tractable processes in the infinite-sample limit. A particular choice of costs yields insight about importance sampling methods, which are a classical tool for coalescent sampling probability approximation. These insights reveal that the behaviour of coalescent importance sampling algorithms differs markedly from standard sequential importance samplers, with or without resampling. We conduct a simulation study to verify that our asymptotics are accurate for algorithms with finite (and moderate) sample sizes. Our results also facilitate the a priori optimisation of computational resource allocation for coalescent sequential importance sampling. We do not observe the same behaviour for importance sampling methods under the infinite sites model of mutation, which is regarded as a good and more tractable approximation of finite alleles mutation in most respects.

翻译：溯祖模型是中性进化下潜在谱系树的基础模型，但其采样概率难以计算。近似这些采样概率的方法要么引入偏差，要么无法扩展到大规模样本。我们证明，在无限样本极限下，一类具有循环突变和有限等位基因数的溯祖成本泛函会收敛到可处理的随机过程。特定成本函数的选择为重要性采样方法提供了新的见解，而重要性采样是近似溯祖采样概率的经典工具。这些见解揭示了溯祖重要性采样算法的行为与标准序列重要性采样器（无论是否采用重采样）存在显著差异。我们通过模拟研究验证了我们的渐近理论在有限（及中等）样本量算法中的准确性。我们的结果还有助于先验优化溯祖序列重要性采样的计算资源分配。在无限位点突变模型下，我们未观察到重要性采样方法具有相同的行为特征，尽管该模型在大多数方面被视为有限等位基因突变的一个优良且更易处理的近似。