Herein, the Hidden Markov Model is expanded to allow for Markov chain observations. In particular, the observations are assumed to be a Markov chain whose one step transition probabilities depend upon the hidden Markov chain. An Expectation-Maximization analog to the Baum-Welch algorithm is developed for this more general model to estimate the transition probabilities for both the hidden state and for the observations as well as to estimate the probabilities for the initial joint hidden-state-observation distribution. A believe state or filter recursion to track the hidden state then arises from the calculations of this Expectation-Maximization algorithm. A dynamic programming analog to the Viterbi algorithm is also developed to estimate the most likely sequence of hidden states given the sequence of observations.

翻译：本文将隐马尔可夫模型扩展至允许观测值具有马尔可夫链性质。具体而言，假设观测构成一条马尔可夫链，其一步转移概率依赖于隐马尔可夫链。针对这一更通用的模型，我们开发了Baum-Welch算法的期望最大化（EM）类比方法，用于估计隐状态与观测的转移概率，以及初始联合隐状态-观测分布的概率。该EM算法的计算过程自然衍生出用于跟踪隐状态的置信状态（滤波器）递推关系。同时，本文还建立了Viterbi算法的动态规划类比方法，以在给定观测序列条件下估计最可能的隐状态序列。