Forward–backward algorithm: Difference between revisions

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==Forward probabilities==

The following description will use matrices of probability values ratherinstead thanof probability distributions. However, althoughit inis generalimportant to note that the forward-backward algorithm can generally be applied to both continuous as well asand discrete probability models.

We transform the probability distributions related to a given [[hidden Markov model]] into matrix notation as follows.

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

:<math>

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Notice that we are now using a [[Row and column vectors|column vector]] while the forward probabilities used row vectors. We can then work backwards using:

:<math>

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Notice that the [[transformation matrix]] is also transposed, but in our example the transpose is equal to the original matrix. Performing these calculations and normalizing the results provides:

:<math>

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==External links ==

* [http://www.cs.jhu.edu/~jason/papers/#eisner-2002-tnlp An interactive spreadsheet for teaching the forward–backward algorithm] (spreadsheet and article with step-by-step walk-through)

* [httphttps://www.cs.brown.edu/research/ai/dynamics/tutorial/Documents/HiddenMarkovModels.html Tutorial of hidden Markov models including the forward–backward algorithm]

* [http://code.google.com/p/aima-java/ Collection of AI algorithms implemented in Java] (including HMM and the forward–backward algorithm)