A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition

II.B Elements of an HMM

1. What are the elements of a discrete HMM \(\lambda = (A, B, \pi)\)?
   \(N\) states and \(M\) observation symbols; transition probabilities \(A = \{a_{ij}\}\); emission probabilities \(B = \{b_j(k)\}\), \(b_j(k) = P[v_k \text{ at } t \mid q_t = S_j]\); initial distribution \(\pi_i = P[q_1 = S_i]\).
   • Rabiner, 1989, A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition (Proceedings of the IEEE), II.B Elements of an HMM
2. How does an HMM generate an observation sequence?
   Choose \(q_1\) from \(\pi\); at each time emit \(O_t\) from \(b_{q_t}\), then move to \(q_{t+1}\) using \(a_{q_t \cdot}\), until \(T\) observations. The same procedure models how a given sequence was generated.
   • Rabiner, 1989, A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition (Proceedings of the IEEE), II.B Elements of an HMM