The hidden Markov model (HMM) is a framework for time series analysis widely applied to single molecule experiments. It has traditionally been used to interpret signals generated by systems, such as single molecules, evolving in a discrete state space observed at discrete time levels dictated by the data acquisition rate. Within the HMM framework, originally developed for applications outside the Natural Sciences, such as speech recognition, transitions between states, such as molecular conformational states, are modeled as occurring at the end of each data acquisition period and are described using transition probabilities. Yet, while measurements are often performed at discrete time levels in the Natural Sciences, physical systems evolve in continuous time according to transition rates. It then follows that the modeling assumptions underlying the HMM are justified if the transition rates of a physical process from state to state are small as compared to the data acquisition rate. In other words, HMMs apply to slow kinetics. The problem is, as the transition rates are unknown in principle, it is unclear, a priori, whether the HMM applies to a particular system. For this reason, we must generalize HMMs for physical systems, such as single molecules, as these switch between discrete states in continuous time. We do so by exploiting recent mathematical tools developed in the context of inferring Markov jump processes and propose the hidden Markov jump process (HMJP). We explicitly show in what limit the HMJP reduces to the HMM. Resolving the discrete time discrepancy of the HMM has clear implications: we no longer need to assume that processes, such as molecular events, must occur on timescales slower than data acquisition and can learn transition rates even if these are on the same timescale or otherwise exceed data acquisition rates.

中文翻译：

---

将HMM泛化为连续时间以实现快速动力学：隐藏的Markov跳跃过程

隐马尔可夫模型（HMM）是用于时间序列分析的框架，广泛应用于单分子实验。传统上，它已用于解释由系统（例如单个分子）生成的信号，这些信号在离散状态空间中演化，该状态空间是在数据采集速率指示的离散时间水平上观察到的。在最初为自然科学以外的应用（例如语音识别）开发的HMM框架内，状态（例如分子构象状态）之间的转换被建模为在每个数据采集周期结束时发生，并使用转换概率进行描述。然而，尽管在自然科学中测量通常是在离散的时间水平上进行的，但是物理系统却根据转换速率在连续的时间内演化。然后得出的结论是，如果物理过程从状态到状态的转换速率与数据采集速率相比较小，则可以证明HMM的建模假设是合理的。换句话说，HMM适用于慢动力学。问题是，由于过渡速率原则上是未知的，因此尚不清楚HMM是否适用于特定系统。因此，我们必须对物理系统（例如单个分子）的HMM进行泛化，因为它们会在连续时间内在离散状态之间切换。为此，我们利用了在推断马尔可夫跳跃过程的上下文中开发的最新数学工具，并提出了隐马尔可夫跳跃过程（HMJP）。我们明确显示HMJP降低到HMM的限制。解决HMM的离散时间差异具有明显的含义：