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

$\displaystyle \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

$\displaystyle 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

$\displaystyle \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

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

... et cetera.



Baggenstoss 2017-05-19