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Reestimation of Observation PDF's

In order to update the observation PDF's, it is necessary to maximize

$\displaystyle Q_j = \sum_{t=1}^T w_{tj} \log b_j(O_t). $

over the PDF $b_j$, where

$\displaystyle w_{t,j} = \frac{\displaystyle \alpha_t(j)\; \beta_t(j) }{\displaystyle \sum_{i=1}^{N} \alpha_t(i)\; \beta_t(i) }.$ (13.18)

This is a weighted maximum likelihood (ML) procedure since if $w_{tj} = c_j$, the results are the strict ML estimates. The weights $w_{tj}$ are interpreted as the probability that the Markov chain is in state $j$ at time $t$.