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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</math>

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

:<math>

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</math>

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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</math>

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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Given HMM (just like in [[Viterbi algorithm]]) represented in the [[Python programming language]]:

<syntaxhighlight lang="python">

states = ('"Healthy'", '"Fever'")

end_state = '"E'"

observations = ('"normal'", '"cold'", '"dizzy'")

start_probability = {'"Healthy'": 0.6, '"Fever'": 0.4}

transition_probability = {

'Healthy' "Healthy": {'"Healthy'": 0.69, '"Fever'": 0.3, '"E'": 0.01},

'Fever' "Fever": {'"Healthy'": 0.4, '"Fever'": 0.59, '"E'": 0.01},

}

emission_probability = {

'Healthy' "Healthy": {'"normal'": 0.5, '"cold'": 0.4, '"dizzy'": 0.1},

'Fever' "Fever": {'"normal'": 0.1, '"cold'": 0.3, '"dizzy'": 0.6},

}

</syntaxhighlight>

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<syntaxhighlight lang="python">

def example():

return fwd_bkw(
observations,

states,

start_probability,

transition_probability,

emission_probability,

end_state),

)

</syntaxhighlight>

<syntaxhighlight lang="pycon">

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== References==

{{reflist}}

* [[Lawrence Rabiner|Lawrence R. Rabiner]], A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. ''Proceedings of the [[IEEE]]'', 77 (2), p.&nbsp;257–286, February 1989. [https://dx.doi.org/10.1109/5.18626 10.1109/5.18626]

* {{cite journal |author=Lawrence R. Rabiner, B. H. Juang|title=An introduction to hidden Markov models|journal=IEEE ASSP Magazine |date=January 1986 |pages=4–15}}

* {{cite book | author = Eugene Charniak|title = Statistical Language Learning|publisher = MIT Press| ___location=Cambridge, Massachusetts|year = 1993|isbn=978-0-262-53141-2}}

* <cite id = RussellNorvig10>{{cite book | author = Stuart Russell and Peter Norvig|title = Artificial Intelligence A Modern Approach 3rd Edition|publisher = Pearson Education/Prentice-Hall|___location = Upper Saddle River, New Jersey|year = 2010|isbn=978-0-13-604259-4}}</cite>

==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)

{{DEFAULTSORT:Forward-backward algorithm}}

[[Category:Articles with example Python (programming language) code]]

[[Category:Dynamic programming]]

[[Category:Error detection and correction]]