Forward/Backward Procedure

We wish to compute the probability of observation sequence $ {\bf O}=O_1 O_2\cdots O_T$ given the model $ \Lambda=\{\pi_j, a_{ij}, b_j\}$. The forward procedure for $ p({\bf O}\vert\Lambda)$ is
  1. Initialization:

    $\displaystyle \alpha_1(i) = \pi_i \; b_i(O_1), \;\;\;\; 1\leq i \leq N$ (13.9)

  2. Induction:

    \begin{displaymath}\begin{array}{rcl} \alpha_{t+1}(j) = \left[ {\displaystyle \s...
...{t+1}), && 1\leq t \leq T-1   && 1\leq j \leq N \end{array}\end{displaymath} (13.10)

  3. Termination:

    $\displaystyle p({\bf O}\vert\Lambda) = {\displaystyle \sum_{i=1}^N} \alpha_T(i)$ (13.11)

The backward procedure is
  1. Initialization:

    $\displaystyle \beta_T(i) = 1$ (13.12)

  2. Induction:

    \begin{displaymath}\begin{array}{rcl} \beta_t(i) = {\displaystyle \sum_{j=1}^N} ...
...1}(j), && t=T-1,T-2, \cdots, 1  && 1\leq i \leq N \end{array}\end{displaymath} (13.13)



Baggenstoss 2017-05-19