We propose a unified framework that extends the inference methods for classical hidden Markov models to continuous settings, where both the hidden states and observations occur in continuous time. Two different settings are analyzed: hidden jump process with a finite state space, and hidden diffusion process with a continuous state space. For each setting, we first estimate the hidden states given the observations and model parameters, showing that the posterior distribution of the hidden states can be described by differential equations in continuous time. We then consider the estimation of unknown model parameters, deriving the continuous-time formulas for the expectation-maximization algorithm. We also propose a Monte Carlo method based on the continuous formulation, sampling the posterior distribution of the hidden states and updating the parameter estimation.

翻译：我们提出一个统一框架,将传统隐蔽的马尔科夫模型的推断方法扩大到连续设置,其中隐藏状态和观测都是在连续时间进行的。对两种不同的设置进行了分析:带有有限状态空间的隐藏跳跃过程,以及具有连续状态空间的隐藏扩散过程。我们首先根据观测和模型参数来估计隐藏状态的隐藏状态,表明隐藏状态的后方分布可以连续地用不同的方程描述。然后我们考虑对未知的模型参数的估计,得出预期-最大化算法的连续时间公式。我们还根据连续的配方提出蒙特卡洛方法,对隐藏状态的后方程分布进行取样,并更新参数估计。