Multiple Records

It is fairly straight-forward to extend the Baum-Welch algorithm to the case when multiple observation sequences (“records") are available. Rather than $O_1,O_2,\ldots,O_T$, we have $O^r_1,O^r_2,\ldots,O^r_{T_r}, \;\;\; r=1,2,\ldots,R$. For each record,

1. Run the forward-backward procedure on $O^r_1,O^r_2,\ldots,O^r_{T_r}$ to produce $\alpha^r_t(i)$, $\beta^r_t(i)$,

2. Compute $\xi^r_t(i,j)$, $t=1,\ldots,T_r$ as in (13.14).
3. Compute $\gamma_t^r(i)$ as in (13.15).

Then, we have

$$\hat{\pi}_i = \sum_{r=1}^{R} \gamma^r_1(i) \;\;\;\;\; \hat{a}_{i... ...aystyle \sum_{r=1}^{R} {\displaystyle \sum_{t=1}^{T_r-1}} \gamma^r_t(i) }.$$

Updating the Gaussian mixture parameters requires defining

$$w^r_{t,j} = \frac{\displaystyle \alpha^r_t(j)\; \beta^r_t(j) }{\displaystyle \sum_{i=1}^{N} \alpha^r_t(i)\; \beta^r_t(i) },$$

which leads to $\gamma^r_t(j,k)$ through (13.20). We then have

$$\hat{c}_{jm} = \frac{\displaystyle \sum_{r=1}^{R} {\displaysty... ...splaystyle \sum_{t=1}^{T_r}} {\displaystyle \sum_{l=1}^{M}} \gamma^r_t(j,l) }$$

and

$$\hat{\mbox{\boldmath$\mu$}}_{jm} = \frac{\displaystyle \sum_{r=... ...aystyle \sum_{r=1}^{R} {\displaystyle \sum_{t=1}^{T_r}} \gamma^r_t(j,m), }$$

... et cetera.