Reestimation of HMM Parameters

The reestimation procedure calculates new estimates of $\Lambda$ given the observation sequence ${\bf O}=O_1 O_2\cdots O_T$. We first define

$$\xi_t(i,j) = \frac{\displaystyle \alpha_t(i) \; a_{ij} \; b_j(O_{... ...isplaystyle \sum_{j=1}^N} \alpha_t(i) \; a_{ij} \; b_j(O_{t+1})\beta_{t+1}(j) }$$ (13.14)

and

$$\gamma_t(i) = {\displaystyle \sum_{j=1}^N} \xi_t(i,j).$$ (13.15)

The updated state priors are

$$\hat{\pi}_i = \gamma_1(i).$$ (13.16)

The updated state transition matrix is

$$\hat{a}_{ij} = \frac{\displaystyle {\displaystyle \sum_{t=1}^{T-1}} \xi_t(i,j) }{\displaystyle {\displaystyle \sum_{t=1}^{T-1}} \gamma_t(i) }.$$ (13.17)