The coupled Wright–Fisher diffusion is a multi-dimensional Wright–Fisher diffusion for multi-locus and multi-allelic genetic frequencies, expressed as the strong solution to a system of stochastic differential equations that are coupled in the drift, where the pairwise interaction among loci is modelled by an inter-locus selection. In this paper, an ancestral process, which is dual to the coupled Wright–Fisher diffusion, is derived. The dual process corresponds to the block counting process of coupled ancestral selection graphs, one for each locus. Jumps of the dual process arise from coalescence, mutation, single-branching, which occur at one locus at the time, and double-branching, which occur simultaneously at two loci. The coalescence and mutation rates have the typical structure of the transition rates of the Kingman coalescent process. The single-branching rate not only contains the one-locus selection parameters in a form that generalises the rates of an ancestral selection graph, but it also contains the two-locus selection parameters to include the effect of the pairwise interaction on the single loci. The double-branching rate reflects the particular structure of pairwise selection interactions of the coupled Wright–Fisher diffusion. Moreover, in the special case of two loci, two alleles, with selection and parent independent mutation, the stationary density for the coupled Wright–Fisher diffusion and the transition rates of the dual process are obtained in an explicit form.

耦合 Wright-Fisher 扩散是多位点和多等位基因遗传频率的多维 Wright-Fisher 扩散，表示为在漂移中耦合的随机微分方程系统的强解，其中位点通过位点间选择建模。在本文中，导出了一个与耦合 Wright-Fisher 扩散有关的祖先过程。双重过程对应于耦合祖先选择图的块计数过程，每个轨迹一个。双重过程的跳跃源于合并、突变、单分支，它们发生在一个基因座上，双分支同时发生在两个基因座上。聚结率和突变率具有金曼聚结过程的转变率的典型结构。单分支率不仅包含一种形式的单基因座选择参数，可以概括祖先选择图的速率，而且还包含双基因座选择参数，以包括成对相互作用对单个基因座的影响。双分支率反映了耦合 Wright-Fisher 扩散的成对选择相互作用的特定结构。此外，在具有选择和亲本独立突变的两个位点、两个等位基因的特殊情况下，耦合 Wright-Fisher 扩散的平稳密度和双重过程的转变速率以明确的形式获得。但它还包含双基因座选择参数，以包括成对相互作用对单个基因座的影响。双分支率反映了耦合 Wright-Fisher 扩散的成对选择相互作用的特定结构。此外，在具有选择和亲本独立突变的两个位点、两个等位基因的特殊情况下，耦合 Wright-Fisher 扩散的平稳密度和双重过程的转变速率以明确的形式获得。但它还包含双基因座选择参数，以包括成对相互作用对单个基因座的影响。双分支率反映了耦合 Wright-Fisher 扩散的成对选择相互作用的特定结构。此外，在具有选择和亲本独立突变的两个位点、两个等位基因的特殊情况下，耦合 Wright-Fisher 扩散的平稳密度和双重过程的转变速率以明确的形式获得。