Exploring the state sequence space for hidden Markov and semi-Markov chains

The knowledge of the state sequences that explain a given observed sequence for a known hidden Markovian model is the basis of various methods that may be divided into three categories: (i) enumeration of state sequences; (ii) summary of the possible state sequences in state profiles; (iii) computation of a global measure of the state sequence uncertainty. Concerning the first category, the generalized Viterbi algorithm for computing the top L most probable state sequences and the forward-backward algorithm for sampling state sequences are derived for hidden semi-Markov chains and hidden hybrid models combining Markovian and semi-Markovian states. Concerning the second category, a new type of state (and state change) profiles is proposed. The Viterbi forward–backward algorithm for computing these state profiles is derived for hidden semi-Markov chains and hidden hybrid models combining Markovian and semi-Markovian states. Concerning the third category, an algorithm for computing the entropy of the state sequence that explains an observed sequence is proposed. The complementarity and properties of these methods for exploring the state sequence space (including the classical state profiles computed by the forward–backward algorithm) are investigated and illustrated with examples.