Forward–backward algorithm: Difference between revisions

Revision as of 12:28, 6 August 2019

Revision as of 13:28, 6 August 2019

Line 60:

</math>

We can now make this general procedure specific to our series of observations. Assuming an initial state vector <math>\mathbf{\pi_0pi}_0</math>, (which can be optimized as a parameter through repetitions of the forward-back procedure), we begin with <math>\mathbf{f_{0:0}} = \mathbf{\pi_0pi}_0</math>, then updating the state distribution and weighting by the likelihood of the first observation:

:<math>

\mathbf{f_{0:1}} = \mathbf{\pi_0pi}_0 \mathbf{T} \mathbf{O_{o(1)}}

</math>

Line 75:

:<math>

\mathbf{f_{0:t}}(i) = \mathbf{P}(o_1, o_2, \dots, o_t, X_t=x_i | \mathbf{\pi}_0 )

</math>

Line 87:

:<math>

\mathbf{P}(o_1, o_2, \dots, o_t|\mathbf{\pi}_0) = \prod_{s=1}^t c_s

</math>

Line 95:

\mathbf{\hat{f}_{0:t}}(i) =

\frac{\mathbf{f_{0:t}}(i)}{\prod_{s=1}^t c_s} =

\frac{\mathbf{P}(o_1, o_2, \dots, o_t, X_t=x_i | \mathbf{\pi}_0 )}{\mathbf{P}(o_1, o_2, \dots, o_t|\mathbf{\pi}_0)} =

\mathbf{P}(X_t=x_i | o_1, o_2, \dots, o_t, \mathbf{\pi}_0 )

</math>