Analyses of definitions of Hidden Markov Models

Hidden Markov Models have been widely used, which are usually considered as a set of states with Markovian properties and observations generated independently by those states. This kind of considerations may have confusion and thus needs improvement from the viewpoint of formalization. Moreover, there are different definitions for Hidden Markov Models, the relations between them also needs clarification. Based on different combinations of some probability conditions concerning the current state and the current observation, this paper analyzes several formal definitions and proves their equivalence respectively for one-dimensional and two-dimensional Hidden Markov Models.