Dualities for branching-coalescing processes in population genetics

在数学种群遗传学中，有两种方法被用来理解进化力量对种群的影响。一方面，我们使用随机过程来描述等位基因的频率。另一方面，人们考虑从宗谱考虑中产生的分支合并过程。理想情况下，两种方法之间的联系是通过两个随机过程之间的对偶关系形式化的。与Steven Evans教授一起，我们将开发系统的方法来构建一个有用的分支-凝聚对偶过程的给定类型-频率过程。特别地，我们的目标是识别一类可以通过祖先考虑来分析的随机过程。系统的构建应大大拓宽对偶方法的范围，使其适用于更广泛的模型类别。Wright-Fisher过程是典型的类型-频率过程。在某些特殊情况下，Wright-Fisher扩散的跃迁密度可以用支结对偶过程来表示。这些表示允许有效地模拟扩散过程；这对于从群体遗传数据中推断突变和选择参数具有重要意义。密度表示是从力矩对偶性推导而来的，这只有在特定的例子中才知道。我们将构造我们的对偶过程，使它们能够以一种独立于矩对偶存在的方式来表示跃迁密度。

In mathematical population genetics, two approaches are used to understand the effect of evolutionary forces on a population. On the one hand, one uses stochastic processes describing the allele frequencies forward in time. On the other hand, one considers branching-coalescing processes that arise from genealogical considerations. Ideally, the connection between the two approaches is formalised via a duality relation between two stochastic processes. Together with Prof. Steven Evans, we will develop systematic methods to construct a useful branching-coalescing dual process for a given type-frequency process. In particular, we aim at identifying a class of stochastic processes that can be analysed via ancestral considerations. The systematic construction should considerably widen the scope of duality methods and make them applicable to a wider class of models.Wright-Fisher processes are prominent examples of the classical type-frequency processes. In some special cases, the transition densities of the Wright-Fisher diffusion admits a representation in terms of a branching-coalescing dual process. These representations allow to efficiently simulate the diffusion process; this is important for the inference of mutation and selection parameters from population genetical data. The density representations are derived from a moment duality, which is known only in specific examples. We will construct our dual processes such that they can be used for the representation of the transition densities in a way that is independent of the existence of a moment duality.