A phase-type distribution is the time to absorption in a continuous- or discrete-time Markov chain. Phase-type distributions can be used as a general framework to calculate key properties of the standard coalescent model and many of its extensions. Here, the ‘phases’ in the phase-type distribution correspond to states in the ancestral process. For example, the time to the most recent common ancestor and the total branch length are phase-type distributed. Furthermore, the site frequency spectrum follows a multivariate discrete phase-type distribution and the joint distribution of total branch lengths in the two-locus coalescent-with-recombination model is multivariate phase-type distributed. In general, phase-type distributions provide a powerful mathematical framework for coalescent theory because they are analytically tractable using matrix manipulations. The purpose of this review is to explain the phase-type theory and demonstrate how the theory can be applied to derive basic properties of coalescent models. These properties can then be used to obtain insight into the ancestral process, or they can be applied for statistical inference. In particular, we show the relation between classical first-step analysis of coalescent models and phase-type calculations. We also show how reward transformations in phase-type theory lead to easy calculation of covariances and correlation coefficients between e.g. tree height, tree length, external branch length, and internal branch length. Furthermore, we discuss how these quantities can be used for statistical inference based on estimating equations. Providing an alternative to previous work based on the Laplace transform, we derive likelihoods for small-size coalescent trees based on phase-type theory. Overall, our main aim is to demonstrate that phase-type distributions provide a convenient general set of tools to understand aspects of coalescent models that are otherwise difficult to derive. Throughout the review, we emphasize the versatility of the phase-type framework, which is also illustrated by our accompanying R-code. All our analyses and figures can be reproduced from code available on GitHub.

中文翻译：

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数学群体遗传学中的相型分布：一个新兴框架

阶段型分布是吸收在连续或离散时间马尔可夫链中的时间。相型分布可以用作计算标准聚结模型及其许多扩展的关键属性的通用框架。这里，阶段类型分布中的“阶段”对应于祖先过程中的状态。例如，到最近共同祖先的时间和总分支长度是阶段型分布的。此外，位点频谱遵循多元离散相型分布，并且二轨迹合并重组模型中总分支长度的联合分布是多元相型分布。一般来说，相型分布为聚结理论提供了强大的数学框架，因为它们可以使用矩阵运算进行分析处理。本综述的目的是解释相型理论并演示如何应用该理论来推导聚结模型的基本属性。然后，这些属性可用于深入了解祖先的过程，或者可用于统计推断。特别是，我们展示了聚结模型的经典第一步分析与相型计算之间的关系。我们还展示了阶段型理论中的奖励变换如何轻松计算树高、树长、外部分支长度和内部分支长度之间的协方差和相关系数。此外，我们还讨论了如何将这些量用于基于估计方程的统计推断。为之前基于拉普拉斯变换的工作提供了替代方案，我们基于相型理论推导了小型合并树的可能性。总的来说，我们的主要目的是证明相型分布提供了一套方便的通用工具来理解合并模型的各个方面，否则很难推导。在整个审查过程中，我们强调阶段型框架的多功能性，我们随附的 R 代码也说明了这一点。我们所有的分析和数据都可以从 GitHub 上的代码中复制。