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After a brief discussion of Markov Evolution Formula (MEF) expressed inProbability Bracket Notation (PBN), its close relation with the jointprobability distribution (JPD) of Visible Markov Models (VMM) is demonstratedby introducing Markov State Chain Projector (MSCP). The state basis and theobserved basis are defined in the Sequential Event Space (SES) of Hidden MarkovModels (HMM). The JPD of HMM is derived by using basis transformation in SES.The Viterbi algorithm is revisited and applied to the famous Weather HMMexample, whose node graph and inference results are displayed by using softwarepackage Elvira. In the end, the formulas of VMM, HMM and some factorial HMM(FHMM) are expressed in PBN as instances of dynamic Bayesian Networks (DBN).

Probability Bracket Notation: Markov State Chain Projector, Hidden Markov Models and Dynamic Bayesian Networks