Notes on the EM Algorithm
• N is a positive integer specifying the number of states in the model. Without loss of generality, we will take theN ’th state to be a special state, the final or stopstate. • Σ is a set of output symbols, for example Σ = {a, b} • Θ is a vector of parameters. It contains three types of parameters: – πj for j = 1 . . . N is the probability of choosing state j as an initial state. Note that ∑N j=1 πj = 1. – aj,k for j = 1 . . . (N − 1), k = 1 . . . N , is the probability of transitioning from state j to state k. Note that for allj, ∑N k=1 aj,k = 1. – bj(o) for j = 1 . . . (N − 1), ando ∈ Σ, is the probability of emitting symbol o from statej. Note that for allj, ∑ o∈Σ bj(o) = 1. Thus it can be seen that Θ is a vector ofN + (N − 1)N + (N − 1)|Σ| parameters.