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.