Mathematical models of genetic evolution often come in pairs, connected by a so-called duality relation. The most seminal example are the Wright–Fisher diffusion and the Kingman coalescent, where the former describes the stochastic evolution of neutral allele frequencies in a large population forwards in time, and the latter describes the genetic ancestry of randomly sampled individuals from the population backwards in time. As well as providing a richer description than either model in isolation, duality often yields equations satisfied by quantities of interest. We employ the so-called Bernoulli factory – a celebrated tool in simulation-based computing – to derive duality relations for broad classes of genetics models. As concrete examples, we present Wright–Fisher diffusions with general drift functions, and Allen–Cahn equations with general, nonlinear forcing terms. The drift and forcing functions can be interpreted as the action of frequency-dependent selection. To our knowledge, this work is the first time a connection has been drawn between Bernoulli factories and duality in models of population genetics.

中文翻译：

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伯努利工厂以及群体遗传学赖特-费舍尔和艾伦-卡恩模型中的二元性

遗传进化的数学模型通常成对出现，通过所谓的二元关系连接起来。最具影响力的例子是赖特-费舍尔扩散和金曼合并，前者描述了大群体中中性等位基因频率的随机演化，而后者则描述了从群体中随机抽样个体的遗传祖先。时间。除了提供比任一单独模型更丰富的描述之外，对偶性还常常产生由感兴趣的量满足的方程。我们采用所谓的伯努利工厂（基于模拟的计算中的著名工具）来导出广泛的遗传学模型的对偶关系。作为具体例子，我们提出了具有一般漂移函数的赖特-费舍尔扩散，以及具有一般非线性强迫项的艾伦-卡恩方程。漂移和强迫函数可以解释为频率相关选择的作用。据我们所知，这项工作首次将伯努利工厂与群体遗传学模型的二元性联系起来。