Forward–backward algorithm: Difference between revisions

Revision as of 08:16, 14 December 2013

Revision as of 09:18, 14 December 2013

Line 44:

:<math>\mathbf{P}(O = j)=\sum_{i} \pi_i b_{i,j}</math>

This can be represented in matrix form by multiplying the state vector (<math>\mathbf{\pi}</math>) by an observation matrix (<math>\mathbf{O_j} = \mathrm{diag}(b_{*,jo_j})</math>) containing only diagonal entries. Each entry is the probability of the observed event given each state. Continuing the above example, an observation of event 1 would be:

:<math>\mathbf{O_1} = \begin{pmatrix}

Line 55:

:<math>

\mathbf{f_{0:1}} = \mathbf{\pi} \mathbf{T} \mathbf{O_{o_1}O_1}

</math>

Line 61:

:<math>

\mathbf{f_{0:t}} = \mathbf{f_{0:t-1}} \mathbf{T} \mathbf{O_{o_t}O_t}

</math>

Line 73:

:<math>

\mathbf{\hat{f}_{0:t}} = c_t^{-1}\ \mathbf{\hat{f}_{0:t-1}} \mathbf{T} \mathbf{O_{o_t}O_t}

</math>