Hidden Markov Chains
The idea of the hidden Markov chain was raised by an American mathematician Leonard Baum and his colleagues in the 1960s. The hidden Markov chain is an extension of the Markov chain: compared to the traditional Markov chain, the state $ S_{t} $ of the hidden Markov chain at certain time step $ t $ can not be observed; however, at every time step $ t $, there is an output state $ X_{t} $ and the observable output state $ X_{t} $ only relates to the unobservable state $ S_{t} $. A typical hidden Markov chain model can therefore be illustrated as follows:
