We introduce a systematic method of approximating finite-time transition probabilities for continuous-time insertion-deletion models on sequences. The method uses automata theory to describe the action of an infinitesimal evolutionary generator on a probability distribution over alignments, where both the generator and the alignment distribution can be represented by Pair Hidden Markov Models (Pair HMMs). In general, combining HMMs in this way induces a multiplication of their state spaces; to control this, we introduce a coarse-graining operation to keep the state space at a constant size. This leads naturally to ordinary differential equations for the evolution of the transition probabilities of the approximating Pair HMM. The TKF91 model emerges as an exact solution to these equations for the special case of single-residue indels. For the more general case of multiple-residue indels, the equations can be solved by numerical integration. Using simulated data we show that the resulting distribution over alignments, when compared to previous approximations, is a better fit over a broader range of parameters. We also propose a related approach to develop differential equations for sufficient statistics to estimate the underlying instantaneous indel rates by Expectation-Maximization. Our code and data are available at https://github.com/ihh/trajectory-likelihood

中文翻译：

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有限状态连续时间机的 Indel 演化模型。

我们引入了一种近似序列上连续时间插入删除模型的有限时间转移概率的系统方法。该方法使用自动机理论来描述无穷小进化生成器对对齐概率分布的作用，其中生成器和对齐分布都可以用成对隐马尔可夫模型（成对 HMM）表示。一般来说，以这种方式组合 HMM 会导致其状态空间的倍增；为了控制这一点，我们引入了粗粒度操作来将状态空间保持在恒定大小。这自然会导致近似 Pair HMM 的转移概率演化的常微分方程。TKF91 模型是针对单残基插入缺失特殊情况的这些方程的精确解。对于多残基插入缺失的更一般情况，可以通过数值积分来求解方程。使用模拟数据，我们表明，与之前的近似值相比，所得到的对齐分布更适合更广泛的参数范围。我们还提出了一种相关的方法来开发微分方程以获得足够的统计数据，以通过期望最大化来估计潜在的瞬时插入缺失率。我们的代码和数据可在 https://github.com/ihh/trajectory-likelihood 获取