  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.
