Forward–backward algorithm: Difference between revisions

Revision as of 20:56, 27 September 2016

Revision as of 21:05, 27 September 2016

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This allows us to calculate the new unnormalized probabilities associated with transitioning to a new state vector <math>\mathbf{\pi '}</math> giventhrough Bayes rule, weighting by the likelihood that weeach element of <math>\mathbf{\pi}</math> observegenerated event 1 as:

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The probability vector that results contains entries indicating the unnormalized probability of transitioning to each state and observing the given event. We can now specifymake this processgeneral procedure specific to our series of observations. Assuming that ouran initial state vector is the equilibrium state vector, <math>\mathbf{\pi_{eq}pi}</math>, (which is often the steady-state distribution, if one exists), we begin with:

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\mathbf{f_{0:0}} = \mathbf{\pi_{eq}pi} \mathbf{T} \mathbf{O_t}

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This value is the forward unnormalized probability vector. The i'th entry of this vector provides:

:<math>

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

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