Now that we have defined both the forward and the backward algorithms, we can use them to estimate the structure of the underlying HMM. The procedure is an application of the Expectation-Maximization algorithm, which will be discussed in the next chapter, Chapter 5, EM Algorithm and Applications, and its goal can be summarized as defining how we want to estimate the values of A and B. If we define N(i, j) as the number of transitions from the state i to the state j, and N(i) the total number of transitions from the state i, we can approximate the transition probability P(i → j) with:
In the same way, if we define ...
