Introduction to GHMM

As we have seen, a Hidden Markov Model (HMM) is a Markov chain in which the states are not directly observable. Instead, the output of the current state is observable. The output symbol for each state is randomly chosen from a finite output alphabet according to some probability distribution. A Generalized Hidden Markov Model (GHMM) generalizes the HMM as follows: in a GHMM, the output of a state may not be a single symbol. Instead, the output may be a string of finite length. For a particular current state, the length of the output string as well as the output string itself might be randomly chosen according to some probability distribution. The probability distribution need not be the same for all states. For example, one state might use a weight matrix model for generating the output string, while another might use a HMM. Formally a GHMM is described by a set of four parameters:

• A finite set Q of states.
• Initial state probability distribution $\pi$.
• Transition probabilities T_i,j for $i,j\in Q$.
• Length distributions f of the states (f_q is the length distribution for state q).
• Probabilistic models for each of the states, according to which output strings are generated upon visiting a state.