In order to consider more statistical characteristics, a class of third-order hidden Markov model is proposed. In this model, both state transition and output observation depend on the current state and on the two preceding states as well. Three algorithms of the third-order hidden Markov model are studied and derived, including the forward-backward algorithm for observation sequence evaluation, the Viterbi algorithm for determining the optimal state sequence, and the Baum-Welch algorithm for training the third-order hidden Markov model. A first-order hidden Markov model equivalent to the third-order hidden Markov model is constructed. A theorem of their equivalence is proposed and proved. This study contributes to the algorithmic
theory of the hidden Markov model, and provides a better method to practical applications.