Mastering Machine Learning Algorithms

Forward phase

If we call p_ij the transition probability P(i → j), we define a recursive procedure considering the following probability:

The variable f_tⁱ represents the probability that the HMM is in the state i (at time t) after t observations (from 1 to t). Considering the HMM assumptions, we can state that f_tⁱ depends on all possible f_t-1^j. More precisely, we have:

With this process, we are considering that the HMM can reach any of the states at time t-1 (with the first t-1 observations), and transition to the state i at time t with probability ...