The Weather-Stone Example and the Elvira Software page 21 5. VMM, HMM and FHMM as Dynamic Bayesian Networks page 23 Summary page 26 References page 27 Abstract After a brief discussion of Markov Evolution Formula (MEF) expressed in Probability Bracket Notation (PBN), its close relation with the joint probability distribution (JPD) of Visible Markov Models (VMM) is demonstrated by introducing Markov State Chain Projector (MSCP). The state basis and the observed basis are defined in the Sequential Event Space (SES) of Hidden Markov Models (HMM). The JPD of HMM is derived by using basis transformation in SES. The Viterbi algorithm is revisited and applied to the famous Weather HMM example, whose node graph and inference results are displayed by using software package 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). Dr. Xing M Wang PBN, Markov Time Evolution & HMM Page 1 of 27 2012-12-16 1. Introduction: PBN and Discrete Markov Chain Inspired by the great success of Dirac notation, we have proposed Probability Bracket Notation (PBN) [1], where we have used PBN to discuss Markov chains (see [2] Chap.11). Based on our main topic of this article, we will concentrate on homogeneous, time-discrete first-order Markov chains with finite discrete states.